document resume ed 098 062 evans, diane learning … · cross number puzzle - review of fractions...

DOCUMENT RESUME

ED 098 062 SE 018 191

AUTHOR Evans, DianeTITLE Learning Activity Package, Alg ?bra 103-104, LAPS

23-33.INSTITUTION Ninety Six High School, S. C.PUB DATE [73]NOTE 190p.; See ED 069 504 for related document

EDRS PRICE MF-$0.75 HC-$9.00 PLUS POSTAGEDESCRIPTORS *Algebra; Analytic Geometry; Curriculum;

*Individualized Instruction; *InstructionalMaterials; Mathematics Education; Number Systems;Objectives; Probability; *Secondary SchoolMathematics; Teacher Developed Materials; TeachingGuides; Units of Study (Subject Fields)

IDENTIFIERS *Learning Activity Package

ABSTRACTThis set of 11 teacher-prepared Learning Activity

Packages (LAPS) in intermediate algebra covers number systems;exponents and radicals; polynomials and factoring; rationalexpressions; coordinate geometry; relations, functions, andinequalities; quadratic equations and inequalities; Quadraticfunctions; systems of equations and inequalities; complex numbers;and probability. Each unit contains a rationale for the materialbeing covered; a list of behavioral objectives; a list of resourcesincluding texts (with reading assignments and problem setsspecified), tape recordings, commercial games, filmstrips, andtransparencies; a problem set for student self-evaluation;suggestions for advanced study; and references. (DT)

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Instructionn

I. Road Rationale

II. Read Behavioral Objectivee

III. Resources

A. All work must be done in math notebook withBIOILkaniv

B. Keep your noteboc4 up to date. Your teachermay ask for it at Lny time (without warning) .

C. Work all the EL:c.6it:c* in at least one textfor each objective.

Always check your exercise (see your teacher)

IV. ileltEvaluation

A. Must be taken at completion of activities foreach section.

B. Dona not afrwtt 7r,ur grade in any way.

V. Advanced Study

A. To be done onlx after all previous work hasbeen satisfactorily completed.

B. Must be approved t-y teacher.

VI. Progrees Test and LAP Tc:st

A. Teacher graded

B. Recycling may tOttll place at this time if testis not satisfantor7.

DO NOT LOSE YOUR LAP. If you do, you must buy another one.

SWspy, ItIMMtf.

a

Rationale (The LAP's Purpose)

You have studied merry mathematical systems in

the past. When you first learned to count, you

used the set of natural numbers. In your early

study of arithmetic, you learned how to add, subtract,

multiply, and divide. You soon found that some

division and subtraction problems had no answers. To

handle such situations, the set of integers was dev-

eloped for closure over subtraction, and the set of

.rational numbers was developed for closure over division.

Irrational numbers such as rrand'n 'are not

included in the number sets as yet developed. The set

of real numbers is the union of the rational and irra-

tional numbers. It is the most complete number system

to be developed.

In this LAP you will study the set of real numbers,

its subsets, and properties.

Later, you will extend the field properties of the

Set of real numbers into the field of complex numbers

which give moaning to numerals such as .

1

Section I

Behavioral Objectives

At the completion of your prescribed course of study, you will beable to:

1. Identify the following sets: natural numbers, whole numbers,integers, rational numbers, irrational numbers, and real numberswhen written

a) in set notation

b) es definitions

.2. Determine the relationships between the sets of natural numbers,whole numbers, integers, rational numbers, irrational numbers,and real numbers.

3. Given the following sets: natural numbers, whole numbers, integers,rational numbers, irrational numbers, and real numbers, identifythe properties of each: a) by completing the chart in Appendix I

,b) by answering True or False questionsc) by answering multiple choice questions

4. Given any number system, state whether or not it is a field. Ifit is not, list the missing properties.

5. "liven two or more real numbers, compute their sum, difference,product, and/or quotient.

6. Wri,.1 or select from several definitions, that which defines"density" for a Omen number system.

2

RESOURCES IREADING AND PROBLEMS:

Objective 1

Nizhols, read pp. 1, 14, 21, 33, 34, Ex. 5-8 page 18; 15, 19 page 42.

Filmstrip: Rational and Irrational Numbers

Objective 2

Nichols, read pages 1, 4, 21, 33-34, ex. it -13 pg. 41; 6 page 35.

Objective 3

Vanatta, read pages 5 - 11, 16 - 17, 29 - 37, Ex. 1-16 pages 6 - 7; 1-11page 15; 1-10 page 26; 1-10 page 53.

Nichols, read pages 2-3, 14-17, 17-23, Ex. 1-4 page 2; 3 page 5; 3 page 17;1, 2, 4-10 page 23; 1, 4, 5 page 35; 1, 5, 30-39 pages 37-43.

Dolciani, read pages 11-15, Ex. 1-16 page 12; 1-20 even page 16.

* Appendix I

Objective 4

Nichols, read pages 1, 14, 23, 24, ex. 1 page 17.

Transparencies: T510 Math, Visual Nos. 13, 14 Properties of Real Numbers

iftlleseakC-3453C-3454C-3455C-3456C-3457C-3458C-3459

Objective 5

Vanatta,25.

Nichols,g, h,

Dolciani,

* REQUIRED

Tosching_Tapos:The Commutative PropertyThe Associative PropertyThe Distributive PropertyThe Closure FropertyThe Inverse ElementsThe Real Number SystemIdentity Elements

read pages

read pagesi page 32.

read pages

10-11, 22-23, Ex.

30-31, Ex. 11, 12

22-24, 26-29, Ex.

3

5-14 pages 11-12; 1, 3, 6, 8, 9 page

pages 23-24; 7 page 39; 1 a, b, Co

1-24 even page 26; 1-12 page 29.

RESOURCES I (Cont')

Objective 6

Vanatta, read pages 17-18, Ex. 1, 2, 4-8 page 19; 8 page 35.

Nichols, read pages 27-28, Ex. 1, 3, 4, 5 page 28.

Delciani, read page 256, Ex. 5-16 page 257.

ACTIVITIES:

Complete the chart in Appendix 1.

AUDIO:

For each Wollensak you use, you must secure a worksheet from your teacher, complete

it, and turn in to your teacher.

(Listed on page 3 of this LAP under Objective 4.)

VIDEO:

Filmstrip: Rational and Irrational Numbers

Transparency: T-510, Math, Visual No. 8, Complex Numbers

Transparencies: T-510, Math, Visual Nos. 13, 14 Properties of Real Numbers

GAMES:

Cross Number Puzzle - Review of Fractions II

Cross Number Puzzle - Things to Know About Fractions

The Conversion Game (Bingo game on operations with real numbers)

"Propo" (Bingo game on real number system and properties)

4

Self-Evaluation I

1 I. Identify the following sets of numbers. Write the name of the setin the blank.

2 II. True or False.

11.11

MINIM ImaMIP

10 {See 2, 1, 0, 1, 2 ...}

2. union of rationale and irrationals

3. the natural numbers and zero

4. the set of numbers starting with one and formedby successively adding one

5. all numbers of the formaE where alb c I and b # 0.

6. non-repeating decimals cannot be written as b

7. The real numbers equal the union of the rationals and the integers.

8. The real numbers are a subset of the rational numbers.

9. The integers are a subset of the real numbers.

10. The natural numbers are a subset of the integers.

11. The intersection of the integers and the rationals is 0.

12. The rationals are a subset of the irrationals.

13. The natural nos. are a subset of the whole numbers.

14. The intersection of the rationale and the irrationals is theset of real numbers.

3 III. Multiple choice: for each of the following write the letter(s) forthe correct answer.

15. Which.of these properties do not hold for the reals?

a) closure property for additionb) associative property for multiplicationc) distributive property of multiplication over additiond) additive identitye) none of these

16. Which of these items holds for the natura] numbers?

a) multiplicative inversesb) additive inversesc) multiplicative identityd) additive identity

e) none of these5

BEST CO PY, AVAILABLE

Self-Evaluation (cont')

17. Which of the following does not hold for the rationale?41.0.110.0.

111

IIMPIINOIN111111

a) associative property of multiplicationb) closure for additionc) multiplicative inversesd) commutative for multiplicatione) none of these

18. Which of these holds for the irrationals?

a) multiplicative identityb) additive identityc) multiplicative inversesd) closure :for multiplicatione) none of these

19. Which of these does riot hold for the integers?

a) additive inversesb) multiplicative inversesc) closure for additiond) multiplicative identitye) Pone of these

20. Which of these properties hold for the rationale?

a) additive inversesh) multiplicative identityc) additive identity

d) multiplicative inverses

e)all of these

21. Which of these holds for the real numbers but does not holdfor the irrational numbers?

a) multiplicative inversesb) closure for multiplicationc) additive inversesd) closure for additione) none of these

4 /V, State if each of the following is true or false. If false, statethe property(s) to make the statement true.

0111.11111.111

22. The set of integers is a field.

23. The set of real numbers is a field.

24. The set of irrational numbers 4s a field.

25. The set of natural numbers is a field.

6

Self-Evaluation (cont')

5 V. Multiple Choice: Simplify the following choose the letter of thecorrect answer.

3 1 3 3 726. 3 + a) 5 b) 6 c), 6 d) none of these111111

11

1114.1110ftEmo

27. -3-(-6) is a) -9 b) 3 c)-3 d) -9 e) none of these

-4 -2 -26 -228. 3 + 5 = a) 1 b) 15 c)15 d) none of *hese

3 5 8 8 529. 4 x 3 m a) 12 b) 7 c) 4 d) none of these

3 .5 -3..2

30. -1 * 5 m a) 6 b) 10 c) 10 d) none of these

31. (-8) + (-6) = a) 14 b) -2 c) -14 d) none of these

'32. -9 x -7 = a) -63 b) -16 c) 63 d) 16 e) none of these

33. -81 * 9 o a) 9 b) -9 c) 3 d) none of these

4

34. 9 m: (a) 4.3. b) .49 c) .4 d) none of Vlese

7

35. 26 a( .28 b) .725 c) 7.25 d) none of these

6 VT. Write the definition of density.

APPENDIX 1

Write an X by each property that holds for the given set.Write a circle (0) by each property that does not hold.Do not leave a blank.

PROPERTIES 1NATURAL WHOLE INTEGERS RATIONAL

Closure for +

Closure for x

Commutativefor +

Commutativefor x

Associativefor +

Associativefor x

-.-...

Distributive

Additiveidentity

Additiveinverse

Multiplica-tive identi-ty

Multiplica-t.:.ve inver-

ses

--__

If you have mastered your Behavioral Objectives take the LAP TEST.

8

8isiCOP!

1111111148LE

ADVANCED STUDY

1. Draw a Venn diagram showing the set of real numbersand its subsets.

2. Prove: The sum of two even numbers is an evennumber.

3. Prove: The product of two odd numbers is an oddnumber.

4. Prove: The product of an even and odd number is aneven number.

2a+b5. Prove: VaVb if a < b, then a < 3

6. Find a number for which n2-n+41 is not a prime

number.

7. Prove airrational.

9

a

a

References

1. Vanatta, Glen C. and Goodwin, A. Wilson,Algebra Two, A Modern Course, Charles E.Merrill Publishing Co., 1966.

2. Dolciani, Mary P., Berman, Simon L.,Wooton, William, Modern Algebra Book Two,

3. Nichols, Eugene D., Modern IntermediateAlgebra, Holt, Rinehart and Winston, Inc.,1965.

4. Transparency T-510 Visual No. 8, ComplesNumbers

5. Transparencies T-510 Visuals 13,14 Propertiesof Real Numbers

6. Audio Tapes

Wollensak Teaching Tapes: C-3453C-3454C-3455C-3456C-3457C-3458C-3459

L EARNING

CTIVITY

PACKAGE

Bar COPi. 41/1111,118LE

EXPONENTS AND RADICALS

T"'" I F T.!ED BY LAP NUMBER 24/

WRITTEN BY .AXX414,

4

RATIONALE (The LAP's Purpose)

Have you ever wondered why exponents are added when powers with the

same base are multiplied? Why do we subtract exponents when dividing?

You may have learned how to manipulate exponents and radicals, but do you

really understand what ycu aro doing? If some of these rules have been

forgotten, we will not only recall them in this LAP, but will also learn

why we operate with them as we do.

We will examine operations with exponents and radicals more formally

than in previous studies. Some of the topics covered will be the laws of

exponents, square root, rational number exponents, radical equations, and

scientific notation. Upon completion of this LAP, you should have a good

foundation for future studies in mathematics.

In later LAPS, concepts involving exponents and radicals will be ex-

tended to such important topics as logarithms, rational expressions,

quadratic functions, and complex numbers. Exponents and radicals are the

building blocks for many topics in mathematics and in the applications of

mathematical concepts to scientific studies.

BEST COPY 1111111B

1

*NOTE:

Section 1BEST COPY AVAUBLE


Upon completion of your prescribed course of study, you will be able to:

1. Identify and apply any of the following in simolifyinq, Expressions*

involving integral exponents.

a. Definition of Tntegral Exponent

b. Definition of Zero Exponent

c. Definition of Negative Exponent

d. Product of Powers Property

e. Power of a Power Property

1. Power of a Product Property

g. Power of a Quotient Property

h. Quotient of Powers Property

i. Negative Exponent Property

J, Order of Operations Agreements

2. Given the expression n , identify:

a. the radicalb. the radical signc. the radicandd. the index

3. Given any irrational number, state and/or identify itsapproximate square root.

4. Given any number, compute its square root by using thedivision method.

5. Given any expression involving radicals, write the simplestform* of any of the following

a. any radicalb. a product or quotient of radicalsc. a sum or difference of radicalsd. a product or quotient of sums or differences of radicalse. an expression having a radical as its denominator

6. Simplify radical expressions for which the radicand is arational number.

To simplify a radical or exponential exOession is towrite the expression so that.

1. there is no radical in the denominator2. each exponent of the radicand is a natural number less than the inde3. each exponent is written as a positive exponent4. no real number is expressed in exponential notation5. there are no unnecessary parenthesis

2

APPENDIX I Algebra 103-104 LAP 24

Part I (Objective 3)

Each number below is irrational. Replace each pair of ques-tion marks with a pair of consecutive integers to make a truestatement. They will show the two integers closest to the irrationalnumbers.

Exam01... < V2-0 <

20 is between 4 x 4 and 5 x 5. Hence, 4 < V26 < 5.1.

2.

3.

4.

< < _7.< << \1151 < 7.

< <

Part II (Objective 4)

5.

6.

7.

8.

< s/270< VIC) << N'/TO << Oti <

Below is an example of a square root computation. Use thisexample to find the square root of each of the following.(See the teacher for more explanation)

Example: Find (24384

1 5

V 2 132025 1 43

1 25306 183120 183121

111215

212:3

Thus

6 .1 5

84 .00 00

84

364831

0021

16

157961

0025

1/2 4f5gT x 156.2

Exercise:

Compute the following:

1) rum

2) br3T

3) 1841

4) 116291

5) /46852

Resources

Objective 1

Vanatta, read pp. 8, 243-247, Ex. 1-24 page 244; 1-10 page 247:24-28 page 27.

Oolciani, read pages 117-118, 153-154, 23; Ex. 1-28 even oralpage 120, 1-20 odd written pages 120-121; 1-18 even page 154;17-24 page 156.

BEST CM AVAILABLE

Nichols, read pages 44-45, Ex. 1-3, 5-7 pages 45-47.

Payne, read pages 57-53, 72-76, 348-345 353-354; Ex. 1-48,even pages 58-59; 1-12 page 73; 1-35 even pages 76-77; 1-9page 349; 1 -50 even page 350; 1-21 even pages 354-355.

Pearson, read pages 390-395, Ex. 1-8 pages 331-392; 1-4 page 395.

Introduction to Exonentl(it29r2Pmed) #1A frames 1-14, 22-26,39: #18 frames 105-177: Old frames 80-96: #le 118-128: #1hframes 142-152: #1i frames 178-196: #19 frames 197-204.

Objective 2

Nichols, read pages 47-48, Ex. 4, page 50.

Objective 3

Payne, read pages 5,6; Ex. 19-20 page 6 and Appendix I, part I.

Objective 4

Appendix I, part II.

Otdecc've 5

Vanatta, read pages 248-250, 252-254, 256-257; Ex. 1-9 at topand bottom p. 251; 1-8, 10-15 top page 255; 1-9, 12-16 bottompages 255-256; 1-18 odd page 253; 20,21, 25-27 top p. 255;1-8 page 257.

Dolciani, read pages 258-260, 263-266; E;,. 1-14 even page 261;15-40 odd page 261; 1-12, 16, 19 page 264; 1-12, 36 pages 266-267.

Nichols, read pages 47-48, Ex. 1-30 page 48.

Payne, read pages 5-10, 12-15, 17-18; Ex. 1-18 page 6; 1-36 evenpage 9; 21-48 even, 61-70, 79-82 page 11; 1-26 even, 36-55 evenpage 13; 1-20 odd, 23-28, 39-42 pages 15-16; 31-45 odd pages18-19.

Objective 5

Vanatta, read pages 248-250, Ex. 13, 14, 15, 17, 18, 21 top page 251;10, 13-17 bottom page 251.

Payne, read page 17, Ex. 9-24 page 18.

Self-Evaluation I814,

1. Give a quanified statement for each of the following.

1. The defintion of the zero exponent.

2. The definition of the negative exponent.

3. Product of powers property.

4. Powers of a power property..

5. Power of a product property.

6. Power of a quotient property.

Quotient of powers property.

II. Write in simplest form without negative exponents:

8.

9.

10.

11.

12.

°

.4 = 2 - 9

,3r'sn

a-2'0

2 3 + 7

61.2..Y)3

6a0

b-2

13. ":31:3J6r.

2:t2 3 414. 8x y

2 III. For , complete the following.

ls. P-Ti- is called

16. ,r is called

17. For x , the x is called

18. For x , the n is called

4

5

Self-Evaluation (cone)

11411:04411111184

IV. Write the simplest name for each of the following.

19. Vgannr-se

20. mir

21. h75. 4-22. 431.12 38

23. 10/76- 5)10

24. )64x7 4 2075. la +

26. 3/a + 105 =

3, 3,--27. v54 - il6

28. 5(01 - trir

+ /-29. 1 -

VI. 4- I-3-

30. ;5-- 5-

6 V. simpli:y thu following.

3/ 831. 216

l4ir--32. 9

/57--33. 2

3 --,.34. 2a'

4111

5/77735. 32

3 VI. Give the approximate square root of each of the following.

36. In 37. /T TB 38. 17'0'e-

4 VII. Compute the square root of the following.

39. 17: Or 40. ramm

IF YOU HAVE MASTERED ALL THE BEHAVIORAL OBJECTIVES, TAKE THE PROGRESS TEST.

SECTION 2


Yf di L'A

Upon completing your preslribed counsel of study, you will be &Ws Olt

7, Given a radical expression, write the simplest equivalentexpression using fractional exponents.

8. Given an expression with fractional exponents, write thesimplest equivalent radical expression.

9. Given a real number raised to the power of any rational ntml..cr,write the simplest name for the number.

10. Given an expressionnof the form .rbir , write it in simplest formusing the theorem, tr7 n(v b

11. Given any expression involving rational exponents, simplifyindices, write their product in simplest radical form.

12. Given two or more radical expressions with different indices,write their product in simplest radical form.

13. Given a radical equation, find its solution set.

14. Given a real number, write it in scientific notation.

6

Resources

Objective 7

Dolciani, read pages 333-134, Ex. 1-6 page 335.Payne, read pages 355-356, Ex. 49-62 page 357Nichols, read page 49, Ex. 7 page 50.

Objective 8

Vanatta, read pages 245-246, Ex. 25-30 page 248.Payne, read pages 355-356, Ex. 33-44 page 357.Nichols, read page 49, Ex. 8 par 51.

Objective 9

Vanatta, read pages 245-246, Ex. 2Z-30 page 247; 7-15, 24 page 248.Dociani, read pages 334-335, Ex. 3-10 page 334; 15-20 page 334.Payne, read pages 355-356, Ex. 1-20 even page 357.

Objective 13

Payne, read page 356, Ex. 12, 15, 19, 20 page 357.Introduction to Exr:onents frames 262-271

Objective 11

Vanatta, read pages 245-246; 13-21 page 247.Wooten, 41-46 page 401Pearson, read page 405, Ex. 6-12 page 406Nichols, read pages 4) -50, Ex. 13 page 51

Objective 12

Vanatta, read page 354, Ex. 3, 9, 16, 17, 18 top page 255;10, 17-21, 23 page 25G.:Jearson, read pages 1)6-397, Ex. 1-5 page 398.Wooten, read pages DI-400, Ex. 7-10 page 401

Objective 13

Vanatta, read pages 258-260, Ex. 1-9 page 260Nichols, read pages 51-52, Ex. 1-5 pages 52-53Wooton, read pages 402-403, 334-336, Ex. 27-36 page 332;1-36 pages 336-337.

Objective 14

Nicnols, read page 53, Ex. 1,2,3, page 54.tooton, read pages 412-413, Ex. 1-38 pages 413-414.Payne, read page 351, Ex. 1-16 page 352.Introduction to Exponents frames 51-79

7

Self-Evaluation II

BEVCOPt400844

7 I. For each of the following, write the simplest equivalent expressionusing fractional exponents.

1. A-

8

9

1001.01MOWIMMOMINYMEMO

=1.....1

32. tr-m-q2i

II. For each of the followng write the simplest equivalent expressionusing radicals.

5.

I

(4ab) 3........ am1111......a.prolliOi.

S.

G. 7 5

(3x)1/2

as

411.

3

8.Cra.

III. Write the simplest name for the following.

3

9. 25 2

11.

10. 16 81 9'11.00+41.

5o

164

11. I.

83

3

12. 16 2

313. 814

10 IV, Write the following in simplest form.

14. 72-77-

8

11

12


15. IWIT

16. h1W-0.1.10.111MMINMIIIImalialk

17. %24

V. Write the following in simplest form.2 3

18. sI

. 3

Y Y19. 4

1

20. (64x4):11-.11.11.111.MINMEtimas.1

wamo.

1

21. (-8A6)1

2 2

22. x y

VI. Write the following in simplest radical form.

woo* samiMi111.1.P.mami

OREM. 00..Im 11

411 .

323. (5/-1. ) (2a )

24. (4Ih" ) )

25. t3/5"

26. h ri13 VII. Solve the following radical equations.

2 - 3 m 1

28. 4/X. + 1 . 25

29. y2x + 1 m 411772-1

9

.M.

AV COI0411481x


30. 9- x 2 =5

31. 34-c - 1 wrx-- + 1

14 VIII. Write each of the following in scientific notation.

V

32. 2.61

33. 2,000,600

34. .002712

-235. 30.61 x 10

36. .00027 x 10-6

IF YOU HAVE MASTERED ALL THE BEHAVIORAL OBJECTIVES, TAKE THE LAP TEST.

BEST COP)AVAILABLE

10

BEST COPY AVAILABLE ADVANCED STUDY

I. The following are in depth problems. To get advanced study credit,you must do at least one set and 80%of that set.

1. , Moderr. Algebra. p. 156, numbers 29-38.

2. , tocleznaFttbra. p. 261, numbers 43-50.

3. , Modern Algebra. p. 265, numbers 21,22,24p. 267, numbers 27-30

4. Algebra Two. p. 252, numbers 23-30.

5. Algebra Two. p. 255, numbers 28 30p. 256, number 24p. 257, numbers 11,12

6. Vanatta, Alpiebra Two. p. 260, numbers 13-15

p. 262, numbers 21,22,39,40

II. Develop a game involving exponents.

Dolciani

Dolciani

Dolciani

Vanatta,

Vanatta,

III. Develop a set of rules to adapt the Equation game to operating withexponents.

IV. Can YOU do this?9

The earth gravitational pull on an object is directly proportional to

its mass (m) and inversely proportioned to the square of the distance of

the object from the center of the earth (d-2)

Mathematically this may be written:

F kmd-2

It an ooject experiences a force (weight) of 120 lbs. at the earth's

surface (4 x 103 miles from the center of the earth), what pull or force

would the same cbject experience at 8 x 103 miles from the center of the

earth.

V. EXTRA FOR EXPERTS - Dolciani, Alaebra One, pp. 276-278.

VI. Modern School Mathematics, p. 332 nos. 21-26, page 234 nos. 29-36.

11

IEST COP/ AVAILABLE

REFERENCES

Vanatta (Abbreviation)Vanatta, Glen D., and Goodwin, Wilson A., Algebra Two, A

Modern Course, Charles E. Merrill Publishing Co., 1966.

Dolciani (abbreviation)Dolciani, Mary P., Berman, Simon L., and Wooton, William,

Modern Algebra and211galostrytLoA2122, HoughtonMifflin Co., 1965.

Nichols (abbreviation)Nichols, Eugene D., Modern Intermediate Algebra, Holt,

Rinehart and Winston, Inc., 1965.

Pearson (abbreviation)Pearson, Helen R. and Allen, Frank B., ModernAlgebra,

A LoGical Approach, Book Two, Ginn and Compally, 1966.

Payne(abbreviation)Payne, Joseph N., Zambonl, rloyd F., Lankford, Jr.,

Frances G., Algebra Two, Harcourt, Brace and World, 1969.

Wooton(abbreviation)

Dolciani, Mary P., Wooton, William, Beckenback, Edwin F.,Sharron, Sidney, modern School Mathematics, Algebra 2,Houghton Mifflin Company, 1968.

Odom, Mary Margaret, Introduction to Exponents, A ProgrammedUnit, Holt, Rinehart and 1:inston, Inc., 1964.

L EARNING

CTIVITY

PACKAGE

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41141Z4filt

a. b

POLYNOMIALS AND

FACTOR! NG

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LAP NUMMI

WRITTCN BY Diane Evans

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v;11 u c.t,.tend oor concept of r !,70.hrw al

..Irtco to ihelue the at of polynom:Pl!.. You v:11

lt!rro how to add, subtract, multiply, f:p,40, rFactor plvnvo.iz.1:4. The field propmrfAer. rrn swr,!

oM'onsively J apply inG thlan cpar3tirrq on 1.--7:111-7..

You probably will rccognigo mr.ay n1

ftom vur twevious work in Alr;nhra.

aain only 4n more depth becaoso of thlnr

when wnaing :it'1 r;!tionnl twrossiowl,

:tincLicns, siving

loped la y.)11.: futuru study,

r""

Pa, ic-1.-11# ir s

#

larcollAW034

e

a1.cert.1%; .,%e wialthe or no is it i a prOynoml.A1.*

2. Givon a polynomial:

tatk, the deg,..c:: cf' the polynomial.

b. :state rf It il ei Tolymt11:al over the jnte....era,

or seals.

ntate ir tEnomial trihomAl or Trf^.::5;

qv ,; r

tat:. IA' i !. ;11 701n cr mare variablen.

3. Gli.en a p..:.::;Lot.1 a.a

,t

4. GI% oth ana tw.3 ,

whico represeeta

:.lcze:.t for the variatIon, detern:ne the

1.##M.;! thQ polynomial in it3 rtmlifind form

C:. 4..ux.

C). differeery

o. product. "

d quoticr. fwhero Inc ef hif;her defpne Oivided

by the palynotat.A1 of leaser degree)

5. GiVua any pulyn:=1.11.t :.::m its odditl inverse.

6. Givo any pol;#::::mial, the inv.:Ter; ji

7. Given any To:y omit!. fr...tore over the ratorr.):;, vito !E

us u eactozi:c.a%ic times frInraThe exaxplet 3 art:

Ft:. TeV:usi : !':

Giv.--11 any h L.. Vor , vr! o r i'y l

1.1,.11;1 trian;1;.

* dfi:t! i 1

4-* App'..y 1 11 ; ; vine,r1

BSIcon&mu.; i-)

:!!) page14S Ex. 1-7 tag,. O.

OhJliCTIVE //2

Nienolq, read pp. 59-62, Ex. 1-4 page 62; 4-6 pas: , 91.ieatbon, read pages 153-154, Ex. 1-5 page 155.

61,,1LiliiVE I3

r,lad p. 63, Fx. 1-18 page 63; 1 a-h pngr. 91Vanatta, read pages 56-58, Ex. 4 -46 page 60.r,0!;.!, read pntvv.-, 56-57, Ex. 21-26 page 77.re:c,.5cn, 111 r;k!es 146-141, 8-0 pane r-0.

re.r; pp, 664,8, 70-78, Ex. 1, 2 page 65; 2. 3 page 61;pak;te. (9 /0; 1-2 pages 74-75; 1-ID paco 78,

V:104.-!, nt,:1 61-68, 73, Ex. 1-4, 6, 8, 17, 20, 21. 30 pngen1, 16, 20, 21, 24, 27, 29 pnv 65; 1, 2, 5,

8, 11, 19 pate!; 68-69; 1, 3, 4, 9, 11, 12. 16, 20, 21, 25paite t

favtv, rd ,w iJ4K... '17.61, 57 58, 72-76, Ex. 1-7, 11. 13-17, 23-28;.ages i-n odd page 59; 1-12 page 73, 1-35 even page 76.

Wv.tcn, rual 18-C9, 251-254, 256-258, 272-275; Z. 5-38 evenpag.i 60-61; 1-?6 even pp. 258-259; 1-20 add page 276;

(83.11(1 EVE:

Nichols, road ..)JeOfs 65-66, Ex. 1 page 67; 2 page 01.

OBJECTIVL

it-90, Ex. 1, 2 page 82; 1-2 ;;.Ain, , 84-81; i -2M-o7; 1-2 page 88; 2 page 90.

VanArl, ze,d pages 74-79, Ex. 5, 6, 8, 11, 1.1. I5, 16. ". 5.27. '14, pa it 76; 1, 2, 13, 10, 12, 14. 16, ;7, Id. 21,

1.gt! 79; 1, 5, 7, 13, 16, 10 pagePayne, Leati 1)aget4 65, 63-70, 80-81, Ex. 127 r,,cts. (.4; 1. ?n

odci paw.' 69; 1-30 evt:en page 71; 33-58 (1..1-tpat 61-U.

pae.o:i 165 166, 172-1e4; E7.. 1-5 i!-%1 ron17.-175; 1-2n oda page 176; 13 11-.-!.

04;1:7TM v;

! h9, L. 18 1-.!_;c

:-!0 cuon page 182.

OBDTr. :

re;.1 p14:.:s /40-745, 1, pnr- 7/!rWootoa, L'e.td 614-616, 16 paw, 0!

3

Behavioral Objectival

I. Definition: A polynomial in the variable x in the vet of all

symbols as + a1x + a2x2 + a x where n can be

any non negative integer and the coefficients

ao, a1, an are members of specified set.

Some examples of polynomials in one variable are:

1. 4 - 3x + 6x2 which .can be written 6x2 3x 4.The specified

art, iv the integers. We, therefore, say this is a 221mainx

famiticlaImm.1

1.1x2

2. + 4x + - 5x + 4x42 5

This in a polynomial over the

3. 5y2

- 3y

This is a polynomial over the

4.1

2

This is a polynomial over the

5. 2x

This is a polynomial over theNicho]s

Reft :r to rage 60 of the /text for the definition of a rOynonio.1

in n variables. In this definition, the vo1ho3s

xn represent n different variables nunh yl r: etn.

e

The folloyill3 are exanplea of c:cpl.celr'il V,, ti, Pr-. :C0Z

.1

;)1) 4 2) 4) .1

5:7-1x

QUESTION: In the nunber "0" a C;arlfull dqc.r. it confoll tci

the definition. ?

ANSWER: Yes, it certainly does. Think about it ..... ..!

. . t.

The following are eAam!)Mes 'it.

in Appendix 3. Study thetw aaa 4:1Ln

I t . r t 11 - -0" le in1.0 3.

I. Apply the following patterns whore applicahlr? Ihe

product of two binomials:

a.(x 7)2 x2 y2

b.(x.y)2 x2. 2y y2

c. (x + y) (x - Ar A

(d) (x + y)3 X3 3xv2 V3II. Use the following patLernd where applicable to factor polynomials

(c') (Y3

vp .8, lollop 2 -

over the integers:

2 r,a. ucx %VC 4 Ud)X t 4. (ax 1:; COY + d;

b. ths distributive property to :omovu cr=ln nv..t!:

c. grouping-by-pairs

x24. 2(xy) y2 (x

e. x2- a

2= (x+a) (x-a)

f. x3 + a3 = (x + a) (x2

- ce.7 + a2)

d. + y)

g x3 - a3 = (x a) (x2 + ax + a2)

h. a5+ b5 (a+b) (a" - d3b alb o ahl

i. aS- b5 r: (a-b) (a4 :t2h2 ah7 1 h')

5

Its

A

PRACTICE SuEn t13 At.!,(;:001;11(.

mum 11CNIAL FACTORS

=t'2x2y 4x3y2 + 2gy3. =

* EXERCISr:!:

1. 3ax + 6a"xv

2xy(z + 2x2y + y2) 2. 16x" U.Yy 8xy

=mum OP FORM ax2 + bx + c

EXERCISES:

2x2 + 3x + 3 1. - 2a - 15

(22 3) (x + 1) 2. 10x- Ilx - 6

3. 9a2- 24a 16

III. MaffliargOL2AksilOSUARES

MEW;a. 25x2 . =

(5x - 1) (5x + 1)

b.t2 b2

(t b) (t + b)

C. (a - 49 =

(a + b)2 - 72.=

[(a b) + 71(a 71

(n + b 4. 7) (a 4. - 7)

d. 160 - 1

6

(4t2)2

(4t2 - 1) (4t2 4. 1) ,

[(202 1214t2 1:1,

(2t - 1) (2t I) (4t2 1. 1)

III: THE D1FPE:11E10E OP TV; sti,:tiA.11 '.continuod)

EXERCISES:

1. x2 y27. 9x2 12r, + 43.2 - 16

2. 4a2 - 25b2

3. 9x2 - 49

4. x2 (y 4.'02

5. (2a - b)2 - 25

6. (x2 + 2x + 1) -a.a.

8. x2 + y2 + 2g/ - 9

9. x4 y4

10. 16 - a4 b4

11. 81 - a4

12. b4 - 16a4

IV; POUR TERMS - COMVON EACH PAIR

EKAIEPLE: MERCIISE3:

+ X + y2 + y

+ z) (y2 + Y)

x(y + I) + 7(Y + 1) =

(x + Y) (7 4' 1)

2x - 2

2 re + 2r + 3s + 6

3. 2xy y 6x - 3

4. + 3ab 3b

5. 3ax 5bx 5by2V: I.M.M4S OP THE FORM ax2 b.y....±2,111:1211.0117A74 EQUARES

MIELE: =RC ISF.5

a. 4a2

+ 12ab + 9b2= 1. 92 + 24;ly 4' 16:,2

(2a 4.. 3b) (2a + 313) 2. 4t2 + itt +,(2a + 3b)2 3. 4k2

+ 16km q. 1un

b. 6x2 + 11x7 + 4y2 = 4. 3622 + e:.'r);

(2x + y) (3x + 4y) 5. 61-.2 12y2

6. 20t2 9t,q - 2012

7

VI: SUM OP CU:V.3

FaXprj.:i.ga:EXAMPLI:

a. , a3 + b3 1. z3 1. 125

(a + b) (a2 ab + b2) 2. 27x ; .1- 1

b. t 3 + 8 n J. 64 + y 3

t5 + 23 = 4. x6 + y6

( t 2) (t2 2t + 4) r 27:e° * Co/. -5

VII: JIFFERE:CE OF CUBES,

IVABCPLE: EtERC7S.ES:

a. a3 - b31. 64 - 2703

(a - b) (it2 + ab + b2) 2. x' - 125

b. x3 - 27 r. 3. - 6

X3 33 4. 3437,43 ey 3

(x- 3) .(X2 + 3x + 9) 5. ex3 - 125y 3

VIII. OTHER TYPES - COMBINATIONS - FACTOR USING THE DIAPP7MT PIETHOU

a. COMPLETE FACTOR= b. COMMON BniOKIAL FACTORS

EXERCISES: EXERCISES:

1. 032 - rib2 1. 3(a - b) 4y.(a b)

2. a4 - 1 2. (x2 y2) - 5(x y)

c. GROUPING POLVUO!&IAIS

1. ax + ay - bx - by 1. 012 4 9b2 - 25C6 - 1210

2. a 3 - a - a2 + 1 2. 25c2

4a4 " 2

- 12011

8

ii

414$141.44044

IiLi-EV ALUATI014

1. IdeAtify which e the tollin ret n polynomial.Write !MS 43:: N9. 11 VP,1.

2.

3.

x2 + 2x + 1

x +1Y.

x2 1. 2x 4.--45 . 6

5. 4 7

2 II. Consider the following polynomial*:

6. 2xy

75

1. --x y

2+ 3x

2+ 2

8. 4/1- x2y

4. x2

+ 2xv + y2

10. I S x2+73 + 3xy + 7.a. Give the degree ux each of the above polynomials.

b. Which of the above polynomials are over the reale,

over the rationale, over the integoro

c. Which of the above are monomials, binomiqint trinomials,

perfect oquare trinomials ?*** -*

I:l. Given the polynomials and replacements below, comrste the valuoof the polynomial.

11. x; x= 2

12. 2(xy - 3); x = -3 and y = 1

13. (1XY2) e(x y)7; x

14. xy + ; x = 1, y = 2, c = -I, d 4

stty.viAuven011(c!wftt9)

Aftteat111. Porten the ladice.tad

Is. Owl het - 0) + ts - 4)

16. (to) 11* (Za 3s f 5)

17. IV 7 (yft2) 22. (1:5 3.3 21..) 4. (3 4ft fist)16. (90 op lilt 3) + (s 3) 23. (190 + 26:41, 544) (1644 4' /PO 0° 70119. 3) 24. (20 - 3x 5)(34 + 4n -20. (a 6)2 25. (60 - 19x2 + 2J2: 10) e ",21. (s b) (s b) 26. (3x2 20(30 + 2y)

Y. True os Pala?

27. .42Y11 a- 02

264 .4212 3x 4) .2x2

21. -(1t-r) y

30, 46640 4ab + b2 9) 4a2 + 4ab bt g

U. Taster ova the integers it passible.

31, 3x2 17s 10

32. al 4a 4

33. 330 121x2

34, xy r; 2x - 6

33. it2 b2

36. GO is AO

37. &" rfa3

31. -x2 f 1

39. x2 - y2 6x +

40. x4 16

41. 6 6x y

42. a" b10

43. x4 - 302x2

VII. raster maw the integers with a ratIsaal nonsatial. ?rotor.

444 72 7 + 46. x2

OPLISOILIMORLE

S %?: DEVALUATION (cone)

VIII. Expand the tollowing using Pascal's triangle.

47) (a - b)9

48) (x 2y) 3

49) (2x + y)5

,50) (2a - 3b) 6

IF YOU HAVE SATISFACTORILY COMPLETED YOUR WORK, TAKE THE LAP TEST,CONSULT YOUR TEACHER FIRST.

BEST COX AVAILABLE

ADVANCE!) :;TIIDY

1. Research and 3earn to use synantic

division. Demonstrate your knowledge

by completing either of the following

groups of exercises:

a. Vanatta, Alspka Two,pp. 71-72,numbers 1,3,6,8, and 10.

b. Dolciani, Modern Algebra, BoolzTwo, p. 523, numbers 1,2,3,4,6.

2. Dolicani, Modern Algebra Two, p. 145

numbers 23-26.

1141011441484

REFERLNCES

Vanatta (abbreviatiol)

Vanatta, Glen D., Ais.hl)raTwo:AlicrnCourscl,Charles E. Merrill Publishing Co., 1966.

Dolciani (abbreviation)

Dolciani, Mary P., Berman, Simon L., !Inc: !:ior.ton.

William, Modern Algebra Book. T -1, Houghton

Mifflin Co., 1965.

Nichols (abbreviation)

Nichols, Eugene D., Modern Tntnrmedinte Algebra,Holt, Rinehart and Winston, Inc., 1965.

Wooton, (abbreviation)

Dolciani, Mary I., Wooton, Willia. Bec.',0r:b1:!!1,

Edwin F., Sharron, Sidney, MIdernMathematics, Algebra 2, Houglv:on Mifflin Co.,1968.

Payne (abbreviation)

Payne, Joseph N., Zamboni, Floyd r., Lankford,Jr., Francis G., Algebra Tuo, Ulrconri,Brace and World, 1969.

Pearson (abbreviation)

Pearson, Mien R., nee Allrn./le.-eLr-.1 , Tcr-i 11

Gins aal

Equations by Layr6nn Ulc-r

13

r

LEARNING

ACTI VITY

PACKAGE

(4'o/hey,.

a.*444.144.

aX moo'

ax Aft%iRx

RAT IONAL

EXPRESS IONS

fou111111111011111111

REVIEWED

\12672

LAP NUMBER 2 6

WRITTEN BY Diane Evans

3

Acknowledgement

The administration and staff of

Ninety Six High School gratefully ac-

knowledges the assistance provided by

the staff of Nova High School, Fort

Lauderdale, Florida. We are especi-

ally indebted to Mr. Lawrence G. Insel

and Mr. Laurence R. Wantuck of Nova's

Math Department for permitting us to

use much material developed by them,

some of which has been reproduced in

its original form.

RATIONALE (The LAP'S Purpose)

In the past you have learned to solve equations. You have

also learned that equations are not always in a form which is

easy to work. Many times it is necessary to simplify equations

to get them in a workable form. This LAP is essential to your

future work in solving equations.

In this LAP "algebraic fractions" will also be reviewed

along with the study of the set of rational expressions under

addition and multiplication. The treatment of the operations

is based on the notion of the quantification of variables over

the set of real numbers and the resulting availability of

field properties. Addition and multiplication of rational

expressions is done first by strict application of definitions

so that the underlying principles, when short cuts are used,

will be understood. The concluding section will make use of

rational expressions in problem solving.

1


After the completion of your prescribed course of study, youwill be able to:

1. Given an expression, determine whether or not it is a rational

expression.

2. Given a rational expression, express it in simplified form; i.e.,

express it so that the numerator and denominator have no common

:7actors.

3. Given a rational expression (or a sum, difference, product or

quotient of two rational expressions), find all replacements for

the variables for which the expression(s) is ondefined.

4. Given two or more rational expressions, determine the least

common denominator for thous expressions.

5. Given a rational expression, find

a) its multiplicative inverse (if it exists)

b) its additive inverse

6. Given two rational expressions, express any of the following in

simplified form:

a) the product of the two expressions

b) the quotient of the two expressions

c) the sum of the two expressions

d) the difference of the two expressions

7. Given a complex rational expression, write its simplified form.

8. Solve equations involving rational expressions.

9. Solve any given word problem involving rational expressions.

814 C°114611LABLE

PErOURCES

Objective 1

Nichols, read pages 95-96, Ex. 1-9 oral page 96. Payne, read pages95-96, Ex. 1-12 page 97.

Objective 2

Vanatta, read pages 81-83, Ex. 1-18 even pages 83-84.Oolciani, read pages 158-160, Ex. 1,3,5,11,15,20,22,26, 29-32pages 160-161

Nichols, read pages 96-98, Ex. 1-9 page 98Wooton, read pages 277-279, Ex. 1-36 pages 279-280Payne, read pages 98-99, 102-103. Ex. 1-12 page 100; 1-10 page102; 1-39 even page 103Pearson, read page 187, Ex. 20 page 53; 1 (a-f) page 191; 28a page 196

Objective 3

Vanatta, read page 21, Ex. 2 page 22Dolciani, read pages 157-158, Ex. 1-24 odd page 158Payne, read page 96, Ex. 13-33 page 97Pearson, Ex. 12 page 534

Objective 4

Nichols, read pages 98-106, Ex. 1 (a-j), 2 (a-j) pages 104-105

Objective 5 -- Appendix I

Objective 6

Vanatta, Parts a, b; read pages 87-90, Ex. 2,3,4,8,9,11,13,14,page 80; 2,4,6,7,8,10 page 90; 5,9,11,12 page 91parts c,d; read pages 81-85, Ex. 2,3,5,13,14,16,19,22,23 pages 86-87; 15,16 page 96Dolciani, parts a,b; read pages 161-162, Ex. 3-6, 8,11,15,16,19-21, 23,24,27,28 pages 162-163; 19,32 page 169Parts c,d; read pages 164-165, Ex. 5,8,10,11,12,15,18,25,26,28,31,32,35,38,42 pages 166-167.Nichols, read pages 98-106, Ex. 1 (a-j) pages 99-100; 1(a -j),.2 (a-j) page 102; 1(a-j), 2 (a-j) pages 107-108Payne, part a ; read page 104, Ex. 1-29 page 105Part b; read page 106, Ex. 1-24 page 107Parts C,d; read pages 110-111, Ex. 1-40 even page 112.Pearson, parts a,b; read pages 187-189, Ex. 2(a-h) page 191Parts c,d; read pages 187-189, Ex. 4 (a-h) page 192; 28 (b,c)page 196.

Objective 7

Vanatta, read pages 91-92, Ex. 1,3,4,9,11 pages 92-93; 18 page 96Dolciani, read pages 167-168, Ex. 1,7,9,10,13,14,25,26 pages 168.169Nichols, read pages 108-109, Ex. 1-9 page 109Wooton, read pages 284-286, Ex. 1-62 odd page 286.Payne, read page 113, Ex. 1-20 page 114

3

iivreptioosit

Pearson, read pages 188-189, Ex. 3 (a-f) page 191; 28 (d-f) page 196

Objective 8

Dolciani, read pages 169-170, Ex. 2,5,6,15,17, 18 page 171

Objective 9

Vanatta, Ex. 1,2,5,6,8,9,12,14,16,17 pages 138-139; 2-4 page 141Dolciani, read pages 173-174, Ex. 1, 3-9, 13, 16 pages 174-176Wollensak C-3808 Reading Written problems.

OBa

1

2

SELF-EVALUATION

I. Which of the following are rational expressions? Circle the numberby each rational expression.

1. 5 3)"..241r

2. 1 x2 +2

3.

31C2-72.

6.

7.

u. 2 e.

Ir. Simplify the following rational expressions:

9.r2 82

r2 3rs + 282

10. x2

x7

11. X2 Z2

yx +yz

13. x2 + XV - 2x

x 27:7

u3 v344.

y( 0)

Self - Evaluation (cont.)

3 III. Decide on the replacements for the variables for which the

following expressions are undefined:

4

15. 3x 2

t2 - t

17. 5 * x37 -

12x 218.

27x

19. 3t * 3

4t - t2 It

rv. What is the Least Common Denominator of each pair of expressions?

VO. x x * 2y 2

21.2

2 2 ;x y

611111l°

22.2x

7:716 (x 4)(x - 2)

23. 1.7:72 rgf

24.6x 2

x (x - 7) x3 Y3

6

5

Self-Evaluation (con0)

V. Find the multiplicative and additive invoree of each of the

following expressions; then simplify.

25.2

26. X

2s/.x2

20. 2-x

U

6 VI. Parform the indicated operation.

33. x 4 x 227

34. 6r 2),7 2y

35. 322 + er e r

9112 r2

36. e .

4x y

2x

-33

37. --7------x"+3x +2 x2-1

3x+4+ x-3

38. x`- 16 x2+8x+16

7

29. y

30.+ 1

31. L.

32. ZS..41

39. e z4. .

40. 813 27 F4371717 20 a. - 6

414x2 . 6x 6x3 =272x -3 4x2 - 12x

5x-x2

x2-2x-842. 2

x 4.8x+12 x2-4x-5

43. 5y 3y

44747.,

,c..,,,

7 VII. Simplify the following complex rational. expressions:

44. hz + Z 462

4.3

v. rLIU *MONO N1OWINOW60

4. X.r7

t ti...:. 472 tu +1. 0

45.lu U

8 VIII. Solve the following:

48. 32x " °3- Ax x-3 x + 4

50. 8 6 39 1X. Solve the following.

49.

')x

3

2x+1

5

= 1_3

51. The average of two numbers is 15. nu,' the numbers if the smalleris two-thirds of the larger.

52. One card-sorter can process a deck of punched cards in 30 minutes,while another can sort the deck in 45 minutes. How long would ittake the two sorters together to process the cards?

53. A solution of silver nitrate in %,,tter 127. silver nitrate. Howmany ounces of the compound must be added to 23 ounces of thissolution to produce a 20% solution?

54. Three men receive together $1285 Flom a business venture.4

5

If A'sshare is $25 more than 7. of B's share, and C's share is --- of B's

1share, find the amcr.Ja 0 of money each should receive.

55. The length of a rectangle is two feet longer than its vldth.Find the width if the perimeter of thv rectangle is 144 feet.

GRADE YOUR OWN TEST. If you have satibfactorily completed your work, youmay take the LAP TEST. CONSULT YOUR TEACHER FIRST.

8

Appendix

Objective 5

Example: The additive inverse of x is -x.

The multiplicative inverse of x is x-1 .

Give the additive inverse and multiplicative inverse of the following:

1. 3 6.

2. k 7.

3. m 8.

4. x 9.

5. 3 10.

9

y-x

-k

- -3

7

1481: coitAva

ADVANCED STUDY

1. Make up a game using rational expressions.

At least 80% of any set of the following problemsMUST BE COMPLETED for credit.

2. Allendoerfer, Fundamentals of Freshman Mathematics,

Ex. 1 -20 p. 74.

3. Allendoerfer, Ex. 17-20 p. 78, 15-22 p. 80.

4. Allendoerfer, Ex. 10-20 p. 80.

5. Dolciani, Modern Algebra Two, Lx. 26, 31, 33, 34,

p. 177, and 9, 10, 15, 19 p. 182.

10

ilet494,4444.0

REFERENCES

Vanatta (abbreviation)

Vanatta, Glen D., and Goodwin, Wilson A., Algebra ModernCou13e, Charles E. Merrill Publishing Co., 1966.


Dolciani, Mary P., Berman, Simon L., and Wooton, William, Modernbllebraa_Book Two, Houghton Mifflin Co., 1965.

Nichol..; (abbreviation)

Nichols, Eugene D., Modern Intermediate Algebra, Holt, Rinehartand Winston, Inc., 1965.


Pearson, Helen R., and Allen, Frank B., Modern_Algebrat A Logical.ApproachL Book Two, Ginn and Company, 1966.


Payne, Joseph N. Zamboni, FLoyd F., Lankford, Jr., Francis G.,Algebra Two, Harcourt, Brace and World, 1969.

Wooton (abbreviation)

Dolciani, Mary P., Wooton, William, Beckenbach, Edwin F., Sharron,Sidney, Modern School Mathematics, Algebra 2, Houghton MifflinCompany, 1968.

Allendoefer, C.B., Oakley, C.O., Fundamentals of Freshmen

Mathematics, McGraw-Hill Book Company, Inc., 1959

Wollensak Teaching Tape c-3809 Reading Written Problems.

11

I NT RODUCT ION

TO

COORD 1 NAT E

GEOMETRY

LEARNING

ACTIVITY

PACKAGE

444/494.

REVIEWED BYLAP NUMBER 27

WRITTEN BY Pia), Lk- 6104341

11773 3

RATIONALE (The LAP's.Purpose)

In 1600 European mathematicians worked

with two branches of mathematics - geometry

and algebra. However, there was no link

between these two branches. Rene' Descartes,

a French philoropher and mathematician, pro-

vided the connection in his GeOmetrie,

published in 1637, by devising a scheme for

locating pointn by using numbers. From this

idea the whole subject of analytic geometry

or coordinate geometry has developed.

In this LAP you will begin an introduc-

tion to coordinate geometry. You.will study

the most basic coordinate figure, the straight

line. You will also investigate the concepts

of slope, intercepts, distance, midpoint,

parallelism, and perpendicularity.

SECTION 1

Zehavioral OBjectiyes. 1

At the completion of your prescribed course of study, you will beable.to:

1. Identify or define the following:

a. certesian coordinate syitemb. Descartesc. abscissad. brdihatee. origin

2. Given a coordinate system for a line:

a. Find the coordinate of any given point.b. Given two points, find the coordinate of any point of

the segment joining the two points.c. Given the coordinates of two points, determine the

distance between them.

3. Given a coordinate system for a plane:

a. Given a point, identify its coordinates.b. Given an ordered pair, graph the corresponding point.c. Given a point, identify the quadrant or the axis which

. contains the point..d.. Given the lengths of two sides of a right traingle, use

the Pythagorean Theorem to find the length of the thirdside.

e. Given a geometric figure, one or 'more of whose sideslie along a horizontal or a vertical line, find lengthsof segments or coordinates of points for this figure.

4. Given the coordinates of two points in a plane:

a. Use distance formula to determine the distance betweenthem.

b. Find the coordinates of the midpoint of the segmentjoining them.

c. Find the slope of the line containing them.

5. Given the slope of a line or sufficient information todetermine this slope, decide whether:

a. the line "rises to the right".b. the line "falls to the right".c. the line is horizontald. the line is vertical

2

Obj. 1:

RESOURCES 1

gesto,14404

VA:tette, Algebra Two, read p. 115, 'Ex. Define the terms inObj. One.

Dolciani, Modern Allebre, read p. 81, Ex. Define the terms inObj. One.

Obj. 2: Nichols, read pp. 115-117, 122, ex. 1, 2 page 116; 1-4 pages117-118, 1-6 page 122.

Wooton, read pp. 154-155, 160, ex. 5-8 page 159.Payne, read p. 136, ex. 1-22, pp. 138-139.Pearson, read p. 204, ex. 1-20, p. 205.

Obi. 3: Vanatta, (a) : (b) p. 115-116, ex. 1-4 page 117: (c)-(e)Nichols, read pp. 118-119, ex. 1-9 pages 119-321.Wooton, read pp. 155-157, 160-161, ex. 9-24 page 159.

Pearson, read pp. 205-210, ex. 1-12 pages 210-211.Wollensak Tape C-3852 Graphing Linear FunctionsGames: Graphing Pictures

An Ordered Pair Code

Obj.s 4,5: Vanatta (04a) read pp. 152-151, ex. 1, 2, 4 page 154:(b) read pp. 154-155, ex. 1-3 pages 155-156:(c) read pp. 142-143, ex. 1-12 odd page 145.

(05)

Dolciani, ( #4a,b) read page 294, ex. 1-6 page 295:(c) read pages 84-88, Ex. 1-12 even page 89:

(#5)

Nichols, 04,5 read pages 122-126, 129-133, ex. 1-14 evenpages 122-123; 1-8 pages 124-125; 1-14even page 131; 1-14 even page 133.

Wooton, 04,5 read pages 168-172, 433-437, ex. 1-18 bottompage 173; 1-30 pages 437-438.

Payne, 04,5 read pages 140-146, 148-151, ex. 1-14 page 143;1-12 pages 146-147;18-23 page 153.

Pearson, 04,5 read pages 212-218, ex. 1-4, 12, 13, pages216-217; 1-14 pages 218-219.

Wollensak C-3854 The Slope of a Line

3

SELF-EVALUATION 1

Obie

1 I. a. Define Cartesian coordinate system.

b. For whom is the Cartesian coordinate system named?

c. On the following coordinate plane, label (1) the abscissa,(2) the ordinate, (3) the origin.

2 II. Use this coordinate system to answer the following questions.

(1) Give the coordinate for each of the following points.

A

C

D

(2) Give the distance between the following pairs of points.

A and E

C and D

SELF-EVALUATION 1 (cone)

-.1b III. (1) Uss the following graph and plot these points.

A (2,4)

8 (-6,-4)C (-1,-1)D (0,5)E (-3,4)F (4,-2)

G (5,0)

RUMP um Pllopmmonoum MEMMUMMINIM

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EWAN p mmumoNummilmumpo MUM NIUMMUM WORM ppNEM momum p mme MUM

3c (2) Identify the quadrant or axis which contains each of the followingpoints.

E (-6,8)

F (7, -1)

G (0,-3).

H (-1,-4)

3d (3) Find the length of the third side in each of the following triangles.

A

C

.

B

C=

C

A

(4) Find the length of each of the sides of the following triangle.

111-11111,

5

AC a

CB a

AB al


(3) Write the coordinates for the points of the vertices of the fella/insfigure.

A

B

C

D

"Wwimmim

4 IV. For each of the following pairs of points, find:

(a) the distance between them(b) the coordinate of the midpoint of the segment joining each pair(c) the slope of the line containing eAch pair

(1) (4,0) (0,-3)

(2)(0,5) (-2,-2)

(3) (4,3) (8,7)

(4)(-2,8) (5,-3)

(5)(8,-2) (-3,9)

DISTANCE MIDPOINT SLOPE

S V. Given two ordered pairs: (a) determine the slope of the line joiningthem, and (b) determine if the line: (1) rises to the right

(2) falls to the right(3) is horizontal(4) is vertical

1. (8,2) (3,7)

2. (-2,-5) (-4,-9)

3. (2,7) (9,7)

4. -5,3) (-5,-4)

5. (8,0) (01-2)

A. SLOPE B. DIRECTION

IF YOU HAVE SATISFACTORILY COMPLETED YOUR WORK ON SECTION 1,CONSULT YOUR TEACHER. THEN TAKE THE PROGRESS TEST ON SECTION 1.

6

SECTION 2

lehavioral Objectives

At the completion of your prescribed course of study, you will beable tot

6. Given an equation of a line:

a. write an equivalent equation in slope intercept form.b. determine the slope of the line.c. determine the y - intercept of the line.d. sketch and/or identify the graph of the line.

7. Write the equation of a given line, when given any one of thefollowing:

a. the slope of the line and the y - intercept of the line.b. the coordinates of a point on the line and the slope of the

line.c. the coordinates of two points on the line.

8. Given the coordinates of a point or information sufficient tofind such coordinates, write:

a. the equation of the horizontal line containing this point.b. the equation of the vertical line containing the point.

RESOURCES 2

Obj, 6: Vanatta, read pages 146-148, ex. page 148, work any 10 problems.Nichol's, read page 144, ex. 2 a - j page 144.Wooton, read pp. 161-162, 175-177, ex. 1-12 oral page 162, 5-18

page 172.Payne, read pages 162-165, ex. 22-27 page 166.Pearson, read pages 161-162, 175-177, ex. 1-7 page 222, 4 page 227.Wollensak Tape C-3852 Graphing Linear Functions

C-3855 Slope Intercept FormTransparency 3M The Straight LineGames: Graphing Pictures

An Equation Code

Obj. 7: Dolciani, 7a, read pages 90-93, ex. 19-26 page 93: 7b readpages 90-93, ex. 1-8 page 93: 7c

Nichols, 7a read pages 142-143, ex. 1 page 144: 7b read pp. 142-143, ex. even numbers bottom page 143: 7c 2, 4, 6,8 top page 143.

Wooton, read pages 175-177, Ex. 1-6 and 13-18 orals page 178,1-12 written page 178, 13-24 page 179.

Payne, read pages 162-165, ex. 7a : 7b, 9-17 page 166: 7c,1-8 page 165, 48 page 167.

Pearson, read pages 224-226, Ex. 1 page 228, 1-17 pages 233-236.

Obj. 8: Nichols read pp. 139-140, Ex. 2a,c,e,g,i page 140.Payne, read pp. 162-165, ex. 1-55 even pages 165-167.Pearson, read pp. 223-226, Ex. 1-4 pages 226-227.

7

SELF-EVALUATION 2

Obj.

6 I. Rewrite the following in elope- intercept form, state the slope andy -,intercept, and graph each (use graph paper on next page).

(1) 2x + 3y a -6 M .4110 b a

(2) 2y 0 -4x + 8 m= b

(3) y 0 -x m b

(4) -18x - 6y se 18 m b(5) 3x 6y - 12 m b

7 II. In each problem below, use the given information to write theequation for a line.

.(1) m gm 5, b 2

(2) m is -2, P1(-3,4)

(3) P1(11-1), P2(-19-1)

(4) m T.. -9, b0

(5) m -4

P1(3.-1)

(6) P1(2,3), P

2(-1,-4)

(7) m - I, b se 3

(8) m 5, P1(-2,-4)

(9) P1(4,-3), P2(=6,2)

8 III. A) Write the equation for: (1) the vertical line, and (2) thehorizontal line through (-2,3).

(1) vertical

(2) horizontal

IV. Multiple Choice.

9 1. If a line is vertical and passes through the point (-2,-3) thenits equation is:

(a) x go -3

(b) Y -3(c) x -2

(d) y -2

(e) none of these

8

.4 %Werra.

SELF-EVALUATION 2 (coat')Gra h a er

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SELF - EVALUATION 2 (cont')

2. The line contains (4,-3) and S, x - 2y wA 4has slope h.

3. The line has slope -2 and y - C.intercept (0,-2).

y -x

4. The line contains (1, -1) and D. x + y 0has slope -1.

3. The line has slope h and E. none of thesecontains (2,-1).

WHEN YOU HAVE COMPLETED YOUR RESOURCES AND SELF - EVALUATION,CONSULT YOUR TEACHER. IF YOU HAVE DONE SATISFACTORY WORK, YOU MAYTAKE YOUR PROGRESS TEST ON SECTION 2.

4

SECTION 3

Behsviorai Objectives

satterAwikto


9. Given the slopes of two lines or sufficient information forfinding slopes, determine if,

a. the two lines are parallelb. the two lines are perpendicularc. the two lines are neither parallel nor perpendicular

10. Given the coordinates of points on two lines, such that certainof the coordinates are variables, determine the value of themissing numbers when the lines are:

a. parallelb. perpendicular

11. Given the equation of a line and a point not on the line, writeand/or identify the equation of a line through the given pointand parallel and/or perpendicular to the given line.

RESOURCES 2

Obj. 9: Vanatta, read pp. 150-151, Ex. 3,4 p. 152.Nichols, read pp. 133-135, Ex. 1-3 p. 136.Wooton, read pp. 175-177, 439-440, Ex. 1-8 and 14-18 page 441,

43-44 page 179.Payne, read pages 155-156, Ex. 1-16 pages 157-158.

Obj. 10: Nichols, read pp. 133-135, Ex. 4-9 page 136.Wooton, read pp. 175-177, 439-440, Ex. 41-42 page 179.

Obj. 11: Vanatta, read pp.150-151, Ex. 6-7 p. 152.Wooton, read pp. 175-177, 439-440, Ex. 25-36 page 179, 9-12

page 441.

12

SELF-EVALUATION 3

I. Given the following pairs of slopes, determine if the lines withthese slopes are parallel, perpendicular, or.neither.

2 5

1' I,' 2

3

2. hi, 6

1

.3. 3, -3

4. -5, 2

4 -35. 3,

6. -6, 6

9 II. Given the following pairs ck linear equations, determine if theirgraphs are parallel, perpendicular, or neither.

10

7

7. y 4 x - 17

y 4x + 6

3x - 6y 92x + y 4

2y 3x - 122x + 3y 3

10. 7x + y gm 7

2y -14x + 4

11. Determine x such that the line through P (x,3) andP (-2,1) is parallel to the line through 1P

3(5,-2)

sad P4(1,4).

12. Determine x such that the line through A(x,3) and21(-2,1) is perpendicular to the line throughC(5,-2), and D(1,4) .

13. Determine m such that y trix + 5 is perpendicularto y 2x + 5.

14. Determine C such that Cx + y -2 is parallel to2x + y 6.

13


/V. Write the equation of a line parallel to each given line and throughthe given points.

1. (2,3) y -3x + 1

2. (-3,2) 4x y 6

3. (4,1) y x + 6

4. (-2,-3) 5x +.2y 3

11 V. Write the equation of a line perpendicular to each given line 6through the given points.

1. (3,1) y -x + 2

2. (-1,3) 2x - y 4

3. (-63,-2) 34 - 2y 4

4. (7,-2) 4x 3y to 5

IF YOU HAVE SATISFACTORILY COMPLETED YOUR WORK, CONSULT YOUR TEACHER.THEN TAKE TEE LAP TEST.

14

ADVANCED STUDY

I. Nichols, read pages 137-138, Ex. work any 4 of 1-11, pp. 138-139.

II. Wooton, Ex. 19, 20 page 441.

III. Nichols, Ex. 7, 8 page 129.

IV. Nichols, Ex. 5, 7, 8 page 125.

V. Vanatta, work any 5 of the following: page 152 nos. 8, 9, 10;page 154 nos. 3, 6, 10;page 156 no. 5

VI. Work any 4 of the followiug:

1. Determine an equation of the line satisfying the statedconditions.

4. Through (-3,2) and parallel to the line joining (2,3)and (1,-2).

b. With x - intercept 2 and y - intercept 3.

c. Through (b,-2b) with slope t

2. Show that the figure whose vertices are (2,1), (4,2), (5,2),and (7,3) is a parallelogram.

3. If a line. has x - intercept a (a 0 0) and y intercept b (b 1 0),show that an equation of the line (called the intercept formof the equation) is:

1a

4. If (xl, yi) and (x2, y9) are two points of a line and ifx10 x9, then an equation the line (called the two-point

form of the equation) is:

Y Ylu X (xX1)2 1

5. Show that if the graphs of the linear equations Aix + Bly CIand A2x +

2y C2 are parallel, then A1B2 A2Bit

6. Show that if A213, A11 then the graphs of the linear equationsAix + 0 Cl'afid A2* 4 62 y C2 are either the same line orpArallet lines.

15

REFERENCES

Vanatta, Glen D., and Goodwin, Wilson A., Algebra Two AModern Course, Charles E. Merrill Publishing Co., 1966.

Dolciani, Mary P., Berman, Simon L., and Wooton,Modern Algebra and Trigonometry, Book Two, HoughtonMifflin Co., 1965.

Nichols, Eugene D., Modern Intermediate Algebra, Holt,Rinehart and Winston, Inc., 1965.

Pearson, Helen R. and Alen, Frank 8., Modern Algebra,A Logical Approach, Book Two, Ginn and Company, 1966.

Payne, Joseph N., Zambone, Floyd F., Lankford, Jr.,Frances G., Algebra 3z Harcourt, Brace and World, 1969.

Dolciani, Mary P., Wooton, William, Beckenback, Edwin F.,Sharron, Sidney, Modern School Mathematics, Algebra 2,Houghton Mifflin Company, 1968.

Wollensak Teaching Tapes C-3852C-3855C-3854

3M Transparencies - The Straight Line

L EARNING

ACTIVITY

PACKAGE

RELAT IONS ,

FUNCT IONS,

AND

INEQUALITIES

LAP NUMBER 28

WRITTEN BY if, id &ia.

11873 3

444.

RATIONALE

In mathematics, the concept of a function is very

important and extremely useful. It appears in almost

every branch of the subject. The concept in mathe-

matics, however, has a slightly different meaning than

in ordinary language. We use the word function to de-

note a certain specific type of correspondence between

the elements of two sets. But previous to any discus-

sion on functions one must initially be concerned with

the idea of a relation.

In this LAP emphasis is placed on review and ex-

tension of concepts which are basic to the study of

relations, functions, and inequalities. The basic

definitions important to the study of functions will

be studied more formally than in previous units. You

will have enough experience with graphing and analysis

of graphs so that you should be able to transfer your

knowledge of functions to future studies in mathematics

and to situations in other academic fields or occupa-

tional endeavors.

1

BESTCORAVAILABIE SECTION 1

BEHAVIORAL OBJECTIVES:

At the completion of your prescribed course of study,

you will be able to:

1. Given a pair of sets:

a. Determine the product set (A x B)

b. Construct the graph of the determined product set.

2. Given a set of ordered pairs and a product set,

determine whether or not that set of ordered pairs

in a subset of that product set.

3. Write and/or identify the definition of these terms:

relation, domain, range, and function.

4. Given a relation as a set of ordered pairs, name

its range and domain.

5. Write and/or identify any or all of the five ways to

express a relation.

6. Given a relation, designate it by

(a) a statement

(b) an equition

(c) the roster method (ordered pairs)

(d) constructing a table

(e) displaying its graph

and name its domain and range. Appendix I will be

completed and turned in to the teacher.

7. Given a relation, determine its inverse.

2

.

RESOURCES

Objective 1

Nichols, read pp. 155-157, Ex. 1-16 pp. 157-158.

Pearson, read pp. 31-34, Ex. 1-8 pp. 34-35.

Objective 2

Nichols, read pp. 158-160, Ex. 2 page 160.

Objective 3

Vanatta, read pp. 99-103, Ex. write the definitions of

the words in this goal.

Dolciani, MA, read pp. 203-204, Ex.

Filmstrip: Relations and Functions

Transparency: 3M - Functions

Objective 4

Vanatta, Read pp. 99-103, Ex. 10 page 105.

Dolciani, MA, read pp. 203-204, Ex. 1-8 oral page 205.

Nichols, read pp. 158-160, Ex. 2 page 160.

Wooton, MSM, read pp. 149-151, Ex. 1-10 oral page 152.

Payne, read pp. 196-198, Ex. 1-6 page 197; 1-11, 18-22

pages 198-199.

Objective 5

Vanatta, read pp. 99-103, Ex. write the five ways to expressa relation.

Dolciani, MA, read pp. 203-204, Ex.

Objective 6

Vanatta, read pp. 99-103, Ex. 1-5 pp. 103-105.

Nichols, read pp. 160-164, Ex. 1-3 pp. 163-164.

Wooton, MSM, read pp. 154-158, Ex. 1-24 pp. 158-159.

3

Citel MAIUSti

RESOURCES I (cone)

Payne, read pp. 194-195, Ex. 1-6, 11, 12, 16-20 page 195.

Wollensak teaching tapes - C-3852: Graphing Linear Functions

C-3855: Slope Intercept Form

Objective 7

Nichols, read pp. 164-169, Ex. 1-3 page 168.

Wooton, MSM, read pp. 404-407, Ex. 1-7 (state if inverse

and draw graphs). page 407.

Payne, read pp. 220-222, Ex. 1-5 page 222.

Pearson, read pp. 293-299, Ex. 1-7 pp. 300-301; 3 a h page 306.

4

Objective 1 1.

SELF-EVALUATION 1

Given A =3 (2, 3, 53 and B (3, 5 )1. Find A X B.

2. Graph A X B.

2 Given N = (1, 2, 3, 4, ...). 'Consider N X N. Which of the setsbelow is a relation in N X N?

3 ((0, 1), (1, 0),(2, 3) )

4. ((1, 1), (1, 2), (1, 3))

5. [(i, 2), (2, il, (5, 5),16, 1), (7, 7))

6. C(- 2, 2), (- 6, 2), (5, 5), (6, 6) )

4 III. Identify the domain and range of each of the sets in Part II. Assumethey are relations in A X A.

7. L=

R=

8. D =

Fi =

9. D

R

10. D =

=

5

SELF-EVALUATION 1 (cone) Amp,

.a14 tnit Aitliale

3 IV. Write the definitions of the following words:

1. relation

2. domain

3. range

4.. function

5 V. List the S ways to express a relation.

1.

2.

3.

4.

5.

6 VI.A.GIVEN: the relation 3x + 1 y,

1. write it in words

2. write 5 ordered pairs

3. make a table using these ordered pairsx

4. graph the ordered pairs

5. write the domain

range

6

SELF-EVALUATION 1 (cont')

6 B. Given the relation:(3,9)(-1,-3)(2,6)(0,0)(-3,0)

1. Write an equation

2. Write the relation in words

3. Construct a table

4. Graph the relation

5. Write the domain

range

4 VII. Write the domain and the range of each of the following:

1. (8,1)(7,2)(-3,1)(7,-6)

2. y o x2

3.

0 2

7

5

4. y is equal to twice x

5.

7

DOMAIN RANGE

SELF- EVALUATION 1 (cone)

7 VIII. Write the inverse of the following relations:

(a) i(2,3), (4,4), (61))

(b) x + 2 y

(c) Y 42

(d) ((-1,4), (5,-7), (2,-3), (4,-6))

If you have satisfactorily completed your work, take the PROGRESS TEST.

8

01 010111SECTION 2

BEHAVIORAL OBJECTIVES:

At the completion of your prescribed course of study, you willbe able to:

8. Given a relation, deciae if that relation is a function or not.

9. Given any function, determine the value of the function for

any given number.

10. Given a function, name its inverse and decide if its inverse

is itself a function.

11. Apply the vertical line test to determine if a relation is a

function.

12. Given two functions f and g over the reels and a real number

"a" determine the following:

(a) a f(x) (e) f(g(x))

(b) f(a x) g(f(x))

(c) f(i) + g(x) (g) f(a)

(d) f. (x) g(x)

13. Given a function f, be able to identify it as:

(a) a constant function (d) a linear function

(b) the identity function (e) a non-linear function

(c) the greatest integer function

14. Given a proportion function:

(a) Identify it as a direct proportion function or as an

inverse proportion function.

(b) Find its constant of proportionality (constant of variation)

15. Construct the graph of an inequality of degree 1 (i.e. a relation

which is a subset of the product set R x R).

9

ikltrAtrRESOURCES 2 44144484.

Objective 8

Vanatta, read 105-106, Ex. 3, 5, 7, 8 page 107.

Dolciani, Modern Algebra, read pages 207-208. Ex. 1-24 even pages208-209.

Nichols, read pages 169-173, Ex. 1 page 171.

Payne, read pp. 199-201, 220-222, Ex. 1-3 (checkpoint) and 1-9page 101; 11 page 206; 1-5 page 222.

Wooton, MSM, read pages 154-157, Ex. 5-16 pages 157-158; 13-16page 153.

Pearson, read pages 273-275, Ex. 1, 2, 5 page 276.

Filmstrip: Relations and Functions

Transparencies 3M: Functions

Objective 9

Vanatta, read pages 108-110, Ex. 1-4, 12-15, 20, 26, 28 pages 110-111.

Dolciani, MA, read pages 207-208, Ex. 1-16 page 209.

Payne, read pages 199-201, Ex. 1-5 page 201, 16-21 page 203.

Wooton, MSM, read pages 149-151, Ex. 11-18 oral page 152, 1-8 writtenpage 152.

Pearson, read pp. 277-280, Ex. 1-3 pages 280-281.

Objective 10

Vanatta, read pages 112-113, Ex. 1, 3-8 page 114; 9 page 126.

Nichols, read pp. 169-173, 3x. 3 page 173; 2, 3 page 268.

Payne, read pages 220-222, Ex. 1-20 even page 223; 28-40 pages 224-225; 8-17 and 21-23 page 226.

Pearson, read pp. 293-299, Ex. 1-7 page 300.

Objective 11

Nichols, read pages 169-173, Ex. 2 pages 172-173.

Mt MU NAllitniResources 2 (cone)

Objective 12

Nichols, read pp. 173-176, Ex. 1-7 pages 175-176.

Wooton, MSM, read pp. 152-153, Ex. 1-34 pages 152-153.

Payne, read pp. 215-217, Ex. 1-25 pages 218-219.

Pearson, read pp. 277-280, 288-291, 316-319, Ex. 1-10 pages 280-281;1-7 page 292; 1-10 pages 319-321.

Objective 13

Nichols, read pp. 176-179, Ex. 1-7 page 178.

Wooton, MSM, read pp. 183-183, 187-188, Ex. 1-24 pages 184-185;1-15 page 188.

Payne, read pp. 206-208, Ex. 1-46 pp. 209-212.

Pearson, read pp. 302-305, Ex. 1-2 page 305.

Objective 14


Wooton, MSM, read pp. 180-183, Ex. 1-24 pages 184-185.

Payne, read pp. 206-208, Ex. 1-46 pages 209-212.

Pearson, read pp. 307-308, 310, Ex. 1-14 pages 308-309; 1-13 page 3111.

Filmstrip: Direct Variation

Objective 15

Vanatta, read pp. 121-123, Ex. 2, 3, 6, 7 page 124.


Wooton, MSM, read pp. 164-167, Ex. 1-28 page 167; 1-16 pages 188-189.

Payne, read 441-442, Ex. 1-14 page 442.

Pearson, read pp. 314-315, Ex. 1-7 pages 315-316.

Wollensak teaching tape, C-3806: Inequality and Equality Sentences

Filmstrip: Graphs of Inequalities in One Variable

11

SEW-EVALUATION 2

Obj.

8 I. Determine if each of the following is a function. Write F f6r functionif it is a function. If it is not a function, write R for relation only.

0.1111.11110.111. (4,1)(6,9)(2,1) ( -4,3)

xf.-6 -62. y 8 1 3 7

4. y = x + 1

5. (3,- 3)(4,-4)(5,- 6)(7,- 8)(3, -9)

6.

07 0

3 5

9 4

5 3

9 II. For each of the following functions, find the value indicated.

1. Find f(2) for f(x) = 6x + 1

2. Find f(-3) for f(x) = 2x - 1

3. Find f(0) for f(x) =x + 1

6x

4. Find f(30) for f(x) = x 2 x

8 - 2x

45. Find f(-10) for f(x)

10 III. Write the inverse of each of the following. Stat.,: whether theinverse is a function or only a relation. Circle F or R.

F or R

F or R

1. y-x

I

2.y

7x + 6

-6 -6 -6 -6 -6

4 2 8 9 7

(12)

SELF-EVALUATION 2 (cant')

ForR 3. y 0 x

ForR 4. y 0 -6 - 2x

F or R 5. (4,7)( -3,2)(9,8)(2, -4)(4.8)

F or R 6. x -2 -4 0 10 -3 -2

y 7 3 10 0 8 11

11 IV. Which of the following is not the graph of a function? (Circle thecorrect answer)

a

d.

a.

e.

IP I

C.

12 V. Given f(x) = x2and g(x) = x + 2, find the following:

1) f(x) g(x)

2) f(g(x))

3) g(f(x)) -

4) 2 g(x)

5) g(2x) =

6) f(x) + g(x) =

7) f(100)

13

SELF-EVALUATION 2 (tont')04

13 VI. Classii; the functions below as linear, non-linear constant, greatestinteger, or identity. A function may have two such classifications.

1. f(x) x2

..1111V f(x) - a where a is a real number

3. f(x) 2x

4. f(x) 0 ba

f(x) 0x

14 VII. Identify each of the following functions as direct or inversepropurtional functions; then find the constant of proportionality.

2x1. f(x)=Me.. =....4. 4

14f(x) x

13 VIII. Which of the following when in R x R, are not linear functions?

a. a direct proportion function

b. the greatest integer function

c. the identity function

d. the function defined by y 2x + 3

13,14 IX. Choose from Column B a graph of the type of function g:;:.,en in Column A.

COLUNNA,

32. Inverse roportionfunction

33. Constant function

34. Direct proportionfunction

35. Oreeteet integerfunction

(14)

0.

e.


15 X. Graph the following on the graph paper included. (next page)

(1) y > 3x + 1

(2) y 1 -2x

(3) y < -4x + 1

(4) y -x - 3

If yoll have satisfactorily completed your work, take the LAP test.CONSULT YOUR TEACHER FIRST.

1

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APPENDIX 1

auan AVAILABLE

1. Given the relation: 5 more than twice x is equal to y

a..write an equation

b. write 5 ordered pairs

c. construct a table using the 5 ordered pairs ,y

d. graph the ordered pairs

II. Given the relation :(,g) (VOCA AO (4, -2)("Q, -4)

a. write an equation

b. write the relation in words

c. construct a table

de graph the r elation

III. Given the relation :

a. write an equation

b. write the relaticn in words

c. write the table in ordered pairs

d. graph the relation

11

6

.7111.INNEW

ADVANCED STUDY

I. Newton's Law of Universal Gravitation is expressed as

follows:

G

%F.kgrav)

M mwhere F, . is the attractive,force

R2 gray)

in newtons between two masses (M) and (m). Thole masses are expressed in

kilograms. R is the distance between the two centers of mass and is expressed

in meters. G is the proportionality constant with a value of

G = 6.67 x 10-11

N. m2 /kg 2.

Find the force (P) that the earth with mass M = 5.98 x 10244. exerts on a

body of mass m = 10 kg located at its surface. Radius of earth

R = 6.38 x 106meters

II. Payne, page 225, no. 41

III. Dolciani, page 210, nos. 31-40

IV. Dolciani, read p. 218, ;Ix. 1-16 any 6 page.219

V. Vanatta, page 121 no. 8, page 112 no. 30, page 104 no. 8.

18

REFERENCES


Nichols, Eugene, D., MODERN INTERMEDIATE ALGEBRA,Holt, Rinehart and Winston, Inc., 1965.

Pearson (Abbreviation)

Pearson, Helen R., and Allen, Frank B., MODERN ALGEBRA:A LOGICAL APPROACH, Book 2, Ginn and Company, 1966.


Payne, Joseph N., Zamboni, Floyd F., Lankford, Jr.,Francis, Algebra Two, Harcourt, Brace and World,Inc., 1969.

Wooton (Abbreviation)

Dolciani, Mary P., Wooton, William, Beckenback, Edwin F.,Sharron, Sidney, MODERN SCHOOL MATHEMATICS, ALGEBRA 2,Houghton Mifflin Company, 1968.


Dolciani, Mary P., Berman, Simon L., Freilich, JuliusMODERN ALGEBRA, Book 2, Houghton Mifflin Co., 1965.


Vanatta, Glen D., Goodwin, A. Wilson, Algebra Two, A ModernCourse, Charles E. Merrill Publishing Co., 1966.

Wollensak teaching tapes: C-3806 Inequality and Equalit Sentences

C-3852 Graphing Linear Functions

C-3855 Slope-Intercept Form

Filmstrips: 1. Relations and Functions

2. Direct Variation

3. Graph of Inequalities in One Variable

Transparencies 3M - Functions

LEA li:UNG

ACTI VITY

PACKAGE

QUADRATIC

EQUATIONS

AND

INEQUALITIES

LAP NUMBER 2q

WRITTEN BY Diane Evan6

32973 4

RATIONALE

In the preceding LAP, you studied linear func-

tions which were defined by linear equations in two

variables. Unfortunately, nature was not so kind,

so in your study of science you will need a working

knowledge of all forms of quadratic equations and

inequalities. For example, the cable of the bridge

in the picture above forms a parabolic curve and

can be reduced to a quadratic equation.

In this LAP we will study all quadratic equa-

tions ind inequalities and some other kinds of

equations which aro expressible as quadratic equa-

tions and inequalities.

SECTION 1


NTenAVAILABLE

At the completion of your prescribed course of study, you will be

able to:

1. Define quadratic equation or quadratic function.

2. Sketch or identify the graph of a quadratic function.

3. Given any ;arorable quadratic function, determine its roots.

4. Givcn any qu,lAratic function that is not factorable, determine

its roots in simplest form by couuslasthemaat.

5. State and/or identify the quadratic formula.

6. Gi'ven the equAion Ax2 4 Bx + C = 0, derive the quadratic formula.

7. (,iver auy ivitdkion ihet is not factorable, determine its solutions

by Juldstituting into the quadratic formula.

RESOURCES

Obj I

VanaLa, read p. 16:i, write the definition of quadratic equation.

Nichols, read p. 221, write the definition of quadratic equation,

Uolciani, read p. 220, write the definition.

Payne, read p. 251, write the definition.

Pearson, read p. 337, write the definition.

Obj. 2

Vanatta, read pp. 163-164, ex. 1-10 even page 164.

Dolciani, read p. 220, ex. 5-10 page 234.

Payne, read pp. 251-253, ex. 1-6 page 153.

Pearson, read pp. 337-340, ex. 1 a b, 2, 3 b c e (draw graphs only)

p. 340.

Obj. 3

Vanatta read pp. 165-166, 2, 5, 6, 7, 10, 12, 14, 17, 18 page

167.

Dolciani, ex. 1-8 even, 15, 18, 22, 28 page 136.

2

Resources 1 (Cont')

Wooton, read pp. 265.268, ex. 146 every fourth problem.

Pearson, read pp. 176477, ex. 3 page 178.

Oki. 4

Vanatta, read pp. 167-170, ex. 1, 3, 7, 10, 14 page 172.

Dolciani, read pp. 268-270, 1, 3, 5, 6, 8 written pages 270 -271.

Wooton, read pp. 337-340, ex. 1-8 page 340.

Payne, read pp. 257-259, ex. 1-10 even p. 259; 19-32 odd p. 260.

Pearson, read pp. 355-357, ex. 1-12 even page 183 (solve by completingthe square)

Obj. 5

Vanatta,.read p. 170, state the quadratic formula.

Dolciani, read p. 269, state the quadratic formula.

Nichols, read pp. 224-225, state the quadratic formula.

Wooton, read p. 339, state the formula.

Payne, read pp. 260-261, state the formula.

Obj. 6

Exercise for all books: Derive the quadratic formula.

Vanatta, read pp. 269-170, ex. above.

Dolciani, read p. 268, ex. above.

Nichols, read pp. 224-225, ex. above.

Payne, read p. 260, ex. above.

S-M Transparency 6M - The Quadratic Formula

Obj. 7

Vanatta, read pp. 170-171, ex. 2, 4, 5, 8, 11 p. 172 solve by usingquadratic formula.

Dolciani, read pp. 268270, ex. 10, 13, 15, 18, 20 written p. 270.

Nichols, read pp. 224-226, ex. 2 every other letter page 227.

Wooton, read pp. 337-340, ex. 9-20 even page 341.Payne, read pp. 260-261, ex. 1-20 odd page 262.Pearson, read pp. 355-357, ex. 1, a,b,c,e,g, 3 a,b,c,d page 358.

(3)

SELF -EVAIDATIOi 1

I. Define: quadratic equation.

II. Graph the following.

(A) x2 - 3x - 4 n y

1.........

1.41

I

es II 2

6 0 2

r -i-

I-

_

45

... _ _

Tri i

- .

(b) y 3x2 + 17x'+ 20 (c) y -2x2 «.3x + 2

.1.71......................

........ ...

6

...........

............

..... ...

......................

...............

vmmgw,....1..........

................ 4

2

-2

,............,....1

X..0 ...5 ..4 siee3 2

2--...

4

' .'. 3 4 5 5

.....

..,-.r...1......-.....

......L.

................

r............

...........b...3 :......-...1-

..

.....

.......

y

--..

,

...

....L.

Iii. .1.vv; the fo:Aowing by factoring.

x2-3x+2=0 x

2+103x12b

2=0...7111L

T r 7

/ Al 4

aellIm 11.11 =RENO

n,NUM

......-a.. spol.+8,

dr

o,...\2wimp/

3 ft ...

an.

.

4 .....

III 5 ..1.7 4...anew Is a 40141 III .../1...M. . =Min

/1101.. /.. ../. . MP

6x2-5x+1=0 x -12

IV. Find the roots of each of the following by completing the square.show your work.

1. x2+11x+21.1=0

2o X c, n+ 7-1

4

x-1= 9

2

4 3x2+8 x+2=0

SELF-EVALUATION 1 (cony)

41414V. State the quadratic formula.

VI. Derive the quadratic formula. Begin with Ax2 + Bx + C 0.

VII. Find the rocts of the following by substituting into the quadraticformula. SHOW YOUR WORK!

3r2+r-1=0

( 3 ) 67c2

+10x +3=0

25 . 1 - 7 x =X.

(2) x2 -2x -12:0

00 5x2+x+1=0

If you have satisfactorily completed your work, you may take the progresstest. Consult your teacher first.5

SECTION 2

. Behavioral Objectives


8. Write and/or identify the definition of imaginary numbers.

9. Given any square root you will be able to determine whether it

is imaginary or real. If it is real, you will be able to

determine if it is rational or irrational.

10. Given a quadratic equation ax2 + bx c as 0, a 0 0:.

a. Find the sum of the roots of the equation.SWme b. Find the product of the roots of the equation.

a 11. Given a quadratic equation of the form ax2+ bx + c - 0, a 0 0:

a. Name the discriminant of the equation.

b. Specify the number of real roots of the equation.

c. Give the nature of the roots by determining:

1. If the roots are real or imaginary.

2. If the roots are real determine if they are rational orirrational.

3. If the roots are equal or unequal.

4. How many times the graph will touch the x axis.

12. Write a quadratic equation ax2+ bx + c = 0:

a. Having a given set {r, s} as its solution set.

b. When given the sum and product of its roots.

c. When certain coefficients are unknown and sufficientinformation about the roots of the equaLion is given to findthese coefficients.

13. Given a fractional equation:

a. Write the corresponding quadratic equation. (assuming oneexixts)

b. Find the solution set of the quadratic equation.

RESOUKAS 2

Obj. 8

Vanatta, read page 166, ex. write the definitiOn of itaginary nuMbers.

Payne, read pp. 24-25, ex. write the definition of imaginary numbers.

Obj. 9

Vanatta, read pp. 166, 30-32, ex. Appendix I.

Payne, read pp. 24-25, ex. 1-27 even page 26.

Filmstrip: Rational and Irrational Numbers.

Obj. 10Jai

9_%11Vanatta, read pp. 172-173, ex. 1-10 even pages 173-174.

Dolciani, read page 273, ex. 1-15 odd (oral) page 274.

134 Nichols, read pp. 228, ex. 2 pages 229-230.



Pearson, read pp. 359-362, ex. 2 page 362.

Obj. 11

Vanatta, read p. 175-177, ex. write the discriminant of the equationax2 + bx + c = 0; ex. 1-10 page 177.

Dolciani, read pp. 275-278, ex. write the discriminant of ax2 + bx + c um 0;1-10 page 278.



Obj. 12

Vanatta, read pp. 172-173, ex. 11, 14, 16, 18, 20 page 173. -

* read pp. 273-274, 16-23 even page 274 top; ex. 1, 2,6, 13, 15, 16, 18, 21, 23 bottom pages 274-275.

Nichols, read pp. 228-229, ex. 1, 3-6 pages 'l29-2S0.

Payne, read pp. 246-271, ox. 1-14 pp. 266-267; 1-14 p. 269; 1-18 p. 271.

Wooton, read pp. 343-344, ex. 16-23 p. 345; 1-14 even, 19-24 pages 345-166.

Obj. 13

Nichols, read pp. 230-232, ex. 1, 2 pages 232-233Pearson, read pp. 365, ex. 4, 6 pages 368-369.

* required (7)

SELF -LVALUATION 2

8 I. Write the definition of imaginary numbers.

9 II. State whether each of the folloving is real or imaginary. If real,state if it is rational or irrational.

.1111111111118

01.1101. 0.111=1119

10 III. Write the sum of the roots of the following:

6. y2 - 2y - 9 0

7. 2x2 + 3x 0

8. 2x2 - 8x - 11111110 ..1111.1 s

9. 3x2 + 15x + 2 1.1 0

10 IV. Write the product of the roots of the following:

10. y2 - 2y - 9 is 0

11. 2x2 + 3x 0

12. 2x2 - 8x - 1 0

13. 3x2 + 15x + 2 0

11 V. In each of the following equations:

(a) determine the value of the discriminant.(b) specify the numberof real roots of the equation.(c) determine if the roots are real or imaginary.(d) if.the roots are real, determine if they are rational or irrational.(e) state if the roots are equal or unequal.(f) state how many times the graph touches the x-axis.

14. x2 + 4x + 3 0

8

SELF-EVALUATION 2 (cons')

15. x2

5x + 7 0

16. 3x2 - 2x 4

17. -5x2 x - 3 = 0

13. -3x2 + 4x 4 1 = 0

19. 3x2 - 4x +3

o

12a VI. Write an equation for each of ele following solution sets.

20. 15,-7)

1

21. {0, 3}

22. (- 1 )7' 9

23. + -

f-1 + VT -1 - VI24. t 2 2

12b VII. Given the following sum and product of roots, write an equation.

25. sum = -7 product = 3

26. sum = 9 product = -2

27. sum = product = 0

9


28. SUM 0 product r7

12c VIII. Find the real values to satisfy the conditions given.

29. For what value(s) of "a".will the sum of the rootsbe 8?

x2 (a2 - 2a) x + 3 0

30. For what value(s) of "b" will the equation2x2 + 4x + (2 b b2) - 0 have exactly one root?

13 IX. For each of the following fractional equations:

(a) Write the corresponding quadratic equation (assuming one exists).(b) Find the solution set of the quadratic equation.

1 x - 1 -x

31. 7C. 7 3(): + 2) g° x + Z

32. x - 4 x

If you have satisfactorily completed your work, you may take the PROGRESSTEST. Consult your teacher first.

10

sar 044SECTION 3



14. Given a radical equation:

a. Write the corresponding quadratic equation. (assuming one exists)

b. Find the solution set of the quadratic equation.

c. Find the solution set of the radical equation.

d. State whether the two equations are equivalent.

e. Name the roots of the quc.dratic equation which are not

permissible roots of the radical equation.

15. Given any quadratic inequality:

a. Find the solution set of the inequality.

b. Graph the solution set on a number line.

16. Given a word problem solvable by means of a quadratic equation:

a. Translate .he problem into a quadratic equation.

b. Solve the problem.

17. Given a word problem solvable by mear a fractional equation:

a. Translate the problem into a quadratic equation.

b. Solve the problem.

11

.1

RESOURCES 3

Obj. 14

Work one set of problems.

Vanatta, read pp. 258-260, ex. 1-14 page 260.

Nichols, read pp. 233-235, ex.. 1, 2, pages 235-236.

Payne, read pp. 360-363, ex. 1-26 even pages 361-362; 145 even pp.363-364.

Pearson, read pp. 370-371, ex. 1-3 page 371.

Wooton, read pp. 334-336, ex. 1-36 even page 336.

Dolciani, read pp. 281-282, ex. 1, 3, 13, 15, 17, 24, 26 pages 282-283.

Obj. 15

Vanatta, read pp. 205-206, ex. 1-8 page 208.

Nichols, read pp. 239-243, ex. 1-3 pages 243-245.

Payne, read pp. 276-278, ex. 3-8 (bottom) page 278.

Wooton, read pp. 362-363, ex. 1-16 even page 364.

Dolciani, read pp. 279-280, ex. 1-8 page 280.

Ubj. 16

Vanatta, read pp. 179-181, ex. 1, 2, 4, 7, 15, 19 pages 181-183;13, 14, page 211.

Nichols, read pp. 221-226, ex. 4-12 pageb 227.

Wooton, read pp. 337-340, ex. 1, 2, 7 page 342; ), 2, 4, 7, 13 pages269-270.

Dolciani. _, ex. 32, 38 page 201; 1-4, 8-13 pages 137-138.

Goal 17

Vanatta, read pp. 179-181, ex. 5, 6 page 182; no. 18 page 160.

Nichols, read pp. 230-232, ex. 1, 2 pages 232-233.

Dolciani, read pp. 178-179, ex. 10, 19, page 1C2.

12

.Obj.

SSLImpALUATION 3

14 I. For each of the following radical equations:

a. Write the corresponding quadratic equation (assuming one exists).b. Find the solution set of the quadratic equation.c. Find the solution set of the radical equation.d. State whether the two equations are equivalent.e. Name the roots of the quadratic equation which are not permissible

roots of the .radical equation.

( 1 ) x - 3 3/21

(2) 131 + 1. 3

(.3) sx 4x + 1

t 31-= 4 4' 34 73 7

-15 II. Solve the following inequalities and graph their solution sets onthe real number line.

(5) 3x2 - 5x - 4 < 0

(6) 2x24 5x c 3

13

IN. .

(7) x + 3x 1 10

(8) 3x . x2 t0

SELF. EVALUATION 3 (cone)

lifNO411,...."1111=1..P.M=11111

16 III. Write the equation and solve the following word problems.

(9) Find the 2 consecutive positive integers whose product is 756.

(IC) rind the length of a side of a square if the length of a diagonalis 5 units greater than the length of a side.

(11) The length of a rectangle in 4 feet more than twice the width. If

the area of the rectangle is 30 square feet, find the length andwidth.

(12) The rug in a bedroom is 9 feet by 12 feet. If the area of the rugis 154 sq. feet, how wide is the strip of bare floor around the rug

if the bare strip is of uniform width?

17 N. Write the equation and solve the following.

(13) A certain integer increased by 4 times its reciprocal equals 81/2.

Find the number.


(15) Jim can pick a bushel of apples in 25 minutes. Sam can pick abushel in 15 minutes. How long will it take the boys to pick abushel together?

If you have satisfactorily completed your work, you may take the LAPteat. Consult your teacher first.

15

4

DEBI COPXAPPENDIX I AVAitaLE

State whether each of the following is real or imaginary. If real,state if it is rational or irrational. Write each in simplest form.

1. nu

4. 1-100

5. /X

6. sr

7.

10. A72-3-

ADVANCED STUDY

1. Wooton, page 342 nls. 49, 50

2. Payne p. 264 nos. 55-62

3. Dolciani, p. 271 nos. 49, 50

4. Wooton, page 346, nos. 26-3

5. Wooton, page 342 nos. 49, 50

6. Pearson, page 363, nos. 9-14

7. Dolciani, pages 283-284, nos. 43-46 any 3 problems.

17

4

444'441490


Vanatta, Glen D., and Goodwin, Wilson A., Algebra Two, A ModernCourse, Charles E. Merrill Publishing Co., 1966.


Dolciani, Mary P., Berman, Simon L., and Wooton, William, ModernAlgebra, Book Two, Houghton Mifflin Co., 1965.


Nichols, Eugene D., Modern Intermediate Holt, Rinehartand Winston, Inc., 1965.


Pearson, Helen R., and Allen, Frank B., Modern Al ebra: A LogicalApproach, Book Two, Ginn and Company, 196 .


Payne, Joseph N. Zamboni, Floyd F., Lankford, Jr., Francis G.,Algebra Two, Harcourt, Brace and World, 1969.


Dolciani, Mary P., Wooton, William, Beckenbach, Edwin F., Shnrron,Sidney, Modern School Mathematics, Algebra 2, Houghton MifflinCompany, 1968.

S

4

4

Ellipse

I

1.1

EARNING

CTIVITY

PACKAGE

Parabola

44r.0

4''41,11,04e

Circle

Hyperbola

QUADRATIC FUNCTIONS

ebra 103-I04-.*.morrrerrnmin

REVIEWED BYLAP NI'MBER 3C.)

41073

4

I

RATIONALE

In preceding LAPs you studied functions in general,

and more specifically, the straight* line. Recall that

Descartes is credited with originating the Cartesian

coordinate system. In the concept of coordinates,

Descartes gave mathematicians a new way to look at math-

ematical information. Not only did he show that first

degree, or linear, equations can be graphed as straight

lines, but he also showed that all second degree, or

quadratic equations can be graphed to become circles,

ellipses, parabolas, or hyperbolas. These quadratic

functions are collectively referred to as conic sections.

Conics appear frequently in nature and in numerous

applications; for example, the orbits of planets about

the sun are ellipses. The supporting cables of a sus-

pension bridge form a parabola. The hyperbola appears

as the edges of the shadow cast on a wall by a lamp-

shade. In this LAP we will investigate the graphs of

these quadratic functions in some detail. In addition,

Na will study quadratic inequalities.

SECTION 1


8411Cot40144


Given a relation in R x R, determine whether or not it is a

quadratic function.

2. Write and/or identify the definition of conic section.

3. List and/or identify the four conic sections.

4. Describe and/or identify the descriptions of the following

terms as they relate to a cone:

a. element

b. axis

c. circle

d. ellipse

e. parabola

f. hyperbola

5. Write and/or identify:

a. the definition of a circle

b. the standard form of the equation of a circle with radius r

and center at the origin.

c. The standard form of the equation of a circle with radius r

and center (h,k).

6. Given the equation of a circle, write and/or identify:

a. the center

b. the radius

c. the graph of the circle

7. Given the center and radius of a circle,

a. graph and/or identify the curve

b. write and/or identify the equation in standard form

2

RESOURCES 1

Objective 1

* Nichols, read pp. 195-199, Ex. 1 a-d page 199-200.

Pearson, read pp. 337 -339, Ex. 2 page 340.

Objectives 2, 3, 4

Nichols, read pp. 312-313, Exercise Appendix I.

Wooton, read pp. 456-457, Exercise, Appendix I.

Dolciani, read pp. 330-331, Ex. Appendix I.

Vanatta, read pp. 183-185, Ex. Appendix I.

Pearion, read pp. 697, Ex. Appendix I..

Payne, read page 417, Ex. Appendix Is

Objective 5

Exercise for all books: Write the definition and equationsin Objective 5.

* Vanatta, read pp. 191-192, Ex. above.

Dolciani, read pp. 300-302, Ex. above.

Wooton, read pp. 442-443, Ex. above.

Payne, read pp. 418-419, Ex. above.

3M Transparency: Circle

Objectives 6, 7

* Vanatta, read pp. 191-193, Ex. 1, 3, 5, 6, 8, 10 page 194; 11-16

page 194.

Dolciani, read pp. 300-302, Ex. 11-16 page 302; 1-4, 9, 10 pp.300-

302 (graph and write equation0.

Wooton, read pp. 442-443, Ex. 13-17, 20 page 443; 1-8 page 443 (graph

and write equations).

Payne, read pp. 418-419, Ex. 1-6 page 419; 3, 4, 6, 9-12 page 420.

3M Transparency: Circle

* required

3

I. Tree of False.

SELF-EVALUATION 1

A conic section is the set of pointsplane intersecting a cone.

An axis is a straight line that liessurface of a cone.

determined by a

wholly within the

A parabola is the section of a cone formed by a planethat is perpendicular to one element.

4. A hyperbola is the section of a cone formed by a planethat intersects the cone so that the plane is parallelto one element.

An element is a line that joins the vertex of a conewith the center of the circle that is its base.

6. An ellipse is the section of a cone formed by a planethat cuts completely through the cone perpendicularto the axis. A circle is a special kind of an ellipse.

II. Which of the following are quadratic functions?

0111111111M11

7.

8.

9.

10.

x2 + y 1

y x + 2

3x - 2y2 7

Y a IfiX2 3

III. Match each figure on the left with its name on the right.

4

A. ellipse

B. circle

C. hyperbola

D. parabola

SELF-EVALUATION 1 (emit')

IV. 15. A. Define circle.

B. Write the equation Gf the circle with radius r and center (0,0).

C. Write the equation. of the circle with radius r and center (h,k).

V. rive the center and radius of the following circles and graph each.

16. x2 + Y2 W 36

17, (x + 5)2 + Cy - 8)2 - 4

18. x2 + y2 + 12x + 11 gm 0

19. x2 .1. y2 10x + 4y + 20 0

VI. For each given center and radius (1) graph the curve:, and (2) writethe equation in standard form.

20. C(2,1), r 3 21. C(0,0), r 6

22. C(4,-2), r a a

SELF-EVALUATION 1 (cont.)

2$. C-8,-1), r 3.

11.1MINIMMISI401/1111MINII

If you have satisfactorily completed your work, take the ProgressTest. Consult your teacher first.

SECTION 2


At the completiOn of your prescribed course of study, you will beable tos

S. Write and/or identify the definitions of the following terms:

a. parabola

b. axis of symmetry

c. the value of p

d. focus (F)

e. directrix

f. vertex (V)

9. Write and/or identify a description of the equations x2

4py

and y2

4px.

in. Given any equation of the form x2 4py and/or y

24px;

A. determine the value of p

B. determine the focus

C. determine the equation of the directrix

D. graph the curve, focus, and directrix

11. Giver, a focus and an equation of a directrix,

a. graph the curve

b. write the equation of the parabola

12. Given an equation of the form (y k)2 4p(x - h) or (x h)

2

4p(y - k), determine. the vertex, focus, directrix, and sketch

the graph.

7

RESOUICES 2

BEST COP. AVAILABLE

Objective 8

Exercise for all texts: Write the definitions of the terms inObjective 8.

Vanatta, read pp. 146, 196, Ex. above.

Nichols, read p. 146, Ex. above.



3K Transparency: Parabola

Objective 9

Vanatta, read p. 188, Ex. describe the equations x2

4py and y2

4px

Objective 10

Vanatta, read pp. 188-190, Ex. 1, 2, 5, 7, 9, 10 pages 190-191.

3M Transparency: Parabola

Objective 11

* Vanatta, read pp. 188-190, Ex. 11-16 page 191.

* Dolciani, read page 306, Ex. 15-18 page 306.

Objective 12


Wooton, read pp. 444-448, Ex. 1-6 page 448.

* required

8

SILLF4IVALUATION 2

01.1.

8 I. Define the following:

a. parabola

b. axis of symmetry

c. the value of p

d. Locus (F)

a. directrix

f. vertex (V)

9 IY. Describe the graph of each of the following:

1. x2 4py

2. y2 4px

10 III. Write the value of p for each of these.

1. x2 -16y

2. y2

100x

3. x2

-6y

4. y2

-2x

5. x2 toy

10 IV. For each of the following: (1) determine the value of p. (2) determinethe focus, (3) determine the equation of the directrix, (4) locate twopoints other than the vertex and graph the curve, focus, and directrix.

1. x2

11. 16y

9

2. y2

-20x

3. y2

2x


4. x2

-Sy

11 V. Given the following foci and dirtctrices, graph each curve formed by them.

1. F (2,0) x in -2 2. F (0,-4) y m 4

3.3

F(' ,0)3

1x 2'04. F (0,3)

1y + 0

3

11 VI. Write an equation for each parabola in example VI above.

1.2.

3.

10

4.

44,SELF4VALUATION 2 (cant')

12 VII. For each of the following (1) give the vertex, (2) give the focus,(3) give the directrix, (4) plot the graph, verton focus, and directrix.

(y - 2)2 I6(x + 2)2. (x + 3)

2III -8 Cy 1)

3. (y+ 2)2m -2(x + 1) 4, x

2+ 8x + 8y + 8 0

If you have satisfactorily completed your work, take the PROGRESS TEST.Consult your teacher first.

11

SECTION 3


44,topl

444494


13. Write and/or identify the definitions of the following terms:

A. ellipse4

B. foci

C. vertices

D. major axis, length of major axis

E. minor axis, length of minor axis

14. Given a drawing of an ellipse, identify the following parts:

A. major axis

B. minor axis

C. foci

D. vertices

E. center

15. Write and/or identify the standard form of both an ellipse with

its center at the origin and major axis on the x-axis and an

ellipse with its center at the origin and major axis on the y-axis.

16. Given any equation of an ellipse, determine by looking at the

equation if the major and minor axes are on x or y and/or determine

the length.

17. Given any equation of an ellipse, determine

A. the semi -major (a) and semi-minor (b) axes

B. the foci

C. graph the ellipse, plot the foci and vertices

18. Given the major and/or minor axes, the length of the semi-major (a)

and semi-minor (b) axes and the center at the origin, determine the

equation of the ellipse.

12

OBJECTIVES 3 (cont')

42*19. Given the vertices and foci,

A. determine the equation of the curveOtek

B. 'sketch the curve

20. Given the equation of an ellipse whose center is not at the

origin, determine

A. the center

B. the foci

C. whether the major axis is parallel.to the x or y axis

D. sketch the graph, plot the center, vertices, and foci

RESOURCES

Objectives 13, 14, 111...11

Exercise for all texts: Appendix 2 parts I-IV

Vanatta, read pp. 194-196, Ex. above.



3M Transparency: The Ellipse

Objective 17

* Appendix II part V

Vanatta, read pp. 195 197, Ex. 1, 3, 4, 7, 9 page 197.

Dolciani, read , Ex. 1, 2, 6, 7 page 308; plot foci and vertices.

Wooton, read pp. 449-452, Ex. 1-8 even page 452.

3M Transparency: Thy Ellipse.

Objective 18

Vanatta, read pp. 194-197, Ex. 11, 12 page 198.

Objective 19

Vanatta, read pp. 194-197, Ex. 17, 18 page 198.

Ob ective 20 (work all exercises)

Dolciani, read , Ex. 23, 24 page 309.Vanatta, read pp. 203-204, Ex. 2, 8, 10, 15 page 204.Pearson, Ex. 7 b, e, g page 691.

* required

13

SELF-EVALUATION 3

0117.

13 I. Define the following terms:

a. ellipse

b. foci

c. vertices

d. major axis

e. length of major axis

f. minor axis

g. center

14 II. Using the following graph identify these parts: (a) major axis,(b) minor axis, (c) foci, (d) vertices, (e) center.

15 III. 1. Write the equation of the ellipse with center at the originand major axis on x-axis.

2. Write the equation of the ellipse with center at the originand major axis on y-axis.

14

COEI AVAILABLE SELF-EVALUATION 3(cont')

16 IV. Determine in each of the following if the major axis is on x or yand give its length.

IMIONINISSI

11111111141MIIMIIMI

2 2

I. 16 9 6

x22

2. 100

x2 2

121 + 14-4 is 4

2 2x

4. 3. 49

16 V. Determine in each of the examples in problem V if the minor axis ison x or y and give its length.

3.

4.

17 VI. In each of these find the value of a (the semi-major axis) and thevalue of b (the semi-minor axis).

x2 2

x2 2

1. T.5. 9 1 3. Tr in. 1

2 _214

2_Y

2

2. 100 4' 36 4. 16 ' Vf a 4

17 VII. Find the foci for each of the following.

x2

9

2

3.

x2 +2

/

2

- 1

1. 16 4" 100 64

x2

111

x2

2. T6 + 14.

15

SELF-EVALUATION 3 (coat')

17 Vin. Graph the following llises, plot the foci, and vertices oleo- .

. .

each.

x2 4.a

2.2 2++

'11,

4 x2-er

t00

18 IX. Given the following centers and values of a and b, write an equationfor each ellipse.

1. a.10b . 3

C(4,1)Major axis parallel to x

2. a = 4b 2

C(-1,3)Major axis parallel to y

3. a = 12b = 9

C(0,7)

Major axis parallel to x

4. a - 12b.:. 8

16

C(3,-2)Major axis parallel to y

SELFEVALUATION 3 (coot') ea WY AVAILABLE

19 Z. Given the following vertices and foci, write an equation for eachand sraph the curve, centers are at (0,0).

1. vertices(80)(-8,0) 2. vertices (OM(0,4)

foci (6,0)(-6,0) foci(0,2)(004)

vertices(10,0)(-10,0)

foci(8,0)(-8,0)

4. vertices (0115.(0..!

foci(0,3)(0,3)

20 XI. For each of the following give (1) the center, (2) the foci, (3) tellif the major axis is parallel to x or y, (4) sketch the curve andplot the center, vertices, and foci.

(x + 1)2 + 213_=_21 + 4)

2

1 16 251

2. 100 36 g'

17

1

SELF-EVALUATION 3 (coat')

3. 25x2+ 9y

2- 100x - 36y 89 111 0

If you have satisfactorily completed your work, take the PROGRESSTEST. Consult your tea6her first.

18

SECTION 4


At the completion of your prescribed courss of study, you willbe able to:

21. Write and/or identify the definition of

A. hyperbola

B. transverse axis and its length

C. conjugate axis and its length

22. Given a drawing of a hyperbola, identify the following parts:

A. transverse axis

B. asymptotes

C. conjugate axis

D. foci

E. vertices

F. center

23. Identify the standard form of

A. the ellipse whose center is at the origin and transverse

axis on x

B. the ellipse whose center is at the origin and transverse

axis on y

24. Given the equation of a hyperbola, determine

A. the length of the transverse axis, the length of the conjugate

axis, draw the asymptotes, and sketch the curve

B. the coordinates of the foci and plot them on the graph

25. Determine the equation of a hyperbola when given

A. the transverse axis and the length of a and b

B. the foci and vertices

26. Given any equation of a hyperbola whose center is not the origin,

determine

A. the center

19

SECTION 4

Rift.BEHAVIORAL OBJECTIVES(cone)

"11 Cal afilatb.

B. the length of the transverse and conjugate axes

C. the vertices

D. plot the asymptotes

E. the foci

F. draw and/or identify the sketch, plot the center, vertices,

and foci

RESOURCES

OBJ. 21, 22, 23

Exercise for all resources: Appendix 3

* Vanatta, read pp. 198-201, Ex. above.



3M Transparencies: The Hyperbola

OBJ. 24

* Vanatta, read pp. 198-201, Ex. 1, 2, 4, 7, 9 page 202.

Dolciani, read pp. 311-312, Ex. 1, 2, 4, 7, 10 pages 311-312 (follow

directions in Obj. 24).

Payne, read pp. 427-340, Ex. 3-8 page 430 (follow directions in Obj. 24).

Wooton, read pp. 453-457, Ex. 1-10 even page 457 (follow'directions

in Obj. 24).

3M Transparencies: The Hyperbola.

OW. 25

* Vanatta, read page 201, Lx. 11-14 page 202.

Payne, read pp. 427-430, Ex. 9-12 page 430.

3M Transparencies: The Hyperbola.

20

SECTION 4

RESOURCES (coot')

OW. 26


Wooton, read , Ex. 21, 22 page 458.

Pearson, read , Ex. 7 page 695, (follow directions in Obi. 26).

21 Transparencies: Tha Hyperbola.

* required

21

a

SEP-VALUATION 4

033.

21 I. Writs the definition of hyperbola.

21 II. a. Write the definition of transverse axis, give its length.

b. Write the definition of conjugate axis, give its length.

22 III. Identify these parts of the following graph: (1) transverse axis

(2) asymptotes (3) conjugate axis (4) foci (5) vertices and

(6) center. T -. -14

1111

.7--9itili

.4,,eAin

1111:46 kAi

SISSIINNlimalum

11tionio

4OOIP

Ilaina4 A

An "---"i:.Arar

23 IV. (1) Write the equation of the ellipse whose center is at the originand transverse axis on x.

(2) Write the equation of the ellipse whose center is at the originand transverse axis on y.

24 V. For each of the following (1) determine the length of the transverseaxis and conjugate axis, (2) draw the asymptotes, (3) sketch thecurve, (4) determine the foci and plot them on the graph.

2 21. T8t7 .

(graph is on the following page)

22

SELF-EVALUATION 4 (conr.I )

a ini 1Ili111

1111111111Pi111=11111111EN1110111311111

111111 Ill

MAXIM&

MUM

naliNE.

INN NI111MI

Illa 111111111RN171

ILIDIMELN

_N

Main111111ENNENI all

x

e

IICNNNI

___ - __ _ .-.....-f1. .

...._ ImoI....... =i

i....___1_.+

- -6---.--

-6 - -3 -2

_f_r_.-

2 3

2H_4 6 6

Et

L

,--

---1_.-------1

._ __5

, Li--I-

23

4.

x2

I2


.I;

41=11 WA

Y

0

. 1 OM II IM.... ..1,

41.

111.41Ir

2

-.3

4 6 6

25 VI. Given the following values of a and b and transverse axis, write

an equation for each.

1. a la 2, b le 3, transverse axis on x

2. a m. 4, b 1, transverse axis on y

3. a 2, b = 7, transverse axis on x

4. a 9, b 12, transverse axis on y

25 VII. Given the following foci and vertices, write an equation for

each hyperbola.

1. F(12,0)(-12,0)V (8, 0) (-8,0)

2. F(0,6)(0.-6)V(0,3)(0,-3)

3. F(0,2)(0,-2)V(0,1)(0,-1)

SELF-EVALUATION 4 (cont') BEST COP! AVAILABLE

26 VIII. For each of the following (1) give the center, (2) give the lengthof the transverse and conjugate axes, (3) vertices, (4) foci,(5) plot the asymptotes, Not the curve, center, vertices and foci.

2 2-

311 - SY-4=11 1(1) 4 16

(y + 3) 2 - 5)2

1(2) 64 36

(3) 25x2, y

2+ 150x + 125 0

If you have satisfactorily completedYour work, take the PROGRESS TEST.Consult your teacher first.

25

.7_11

....61_

1

(.1

.110.4011M1

6 -1 3 2 1 1

3--_T--

2 4

6

6

....=11.

-. 7

4

..---- 3

-------,X

2

0 ."-T X7 3 2.1

3 4 6

3.---4

5--- -__., c7

Y"

-6

5

o I7 r, 4 ' 2 1

2I

3I I

1._

1 3 4 6

"-+

t-

SECTION 5

Behavioral ObjectiveslialetrAV411484


27. Given any quadratic inequality, determine its graph.

28. Given any word problem, determine its equation and determine

its solution.

RESOURCES

Obj. 27 (both are required)

Vanatta, read pp. 205-208, Ex. 9-12, 16 page 208.

Dolciani, read , Ex. 10, 12 page 309.

Obj. 28

Vanatta, read pp. 179-181, Ex. 1-4, 6-8, 16, 19 pages 181-182; 13-14

page 211

26

SELF-EVALUATION 5

OW.

27 I. GRAPH the following Inequalities.

(1) x2 + y2 < 36

muummm

liIril=WMgll

ill.=MINN=

momm

II=Ill:

11-g IIoMM x'

III=

l.4

N

..' MINIUMMENNENMUNE

(3) 9y2 x2

--yNN.--- _

1

III

II

MEMINIMM

MIN

M

MINNmamPlis

NENMUNNMEN

NIII

N11M-ENEMra -6 3 -4 -3 -2 2 3 ,

- j

-4-1

I

01

.

27

(2) 9x2+ 25y2 > 225

MMEMMNM

MiiiMBEN

mmsWagamp

MEM

IMPimama

_pm

nom

6

.oNMINN*

M-6-4-f-1-4-_ _,

.41

,_ :4.IMNMMINIM

NNNNNMEINNNIN

ENMN

I.

MINIMmil

'NM MI

(4) x2 g 16y

MMM Y M

MEM

MMMENMMM

*

:11II .

Ems-4 -5 -1 -3 -2 1

laMNN

i

M1NE


28 II. Solve the following problems.

1. Three times the sqearo of a positive integer, decreased by twicethe product of the number and the next smaller integer, is 143.Find the number.

2. Find two consecutive integers such that, if twice the larger isadded to three times the square of the smaller, the sum will be58.

3. The base of a triangle is 4 feet less than the altitude, and thearea of the triangle ib 48 square feet. Find the length of thebase.

4. A rectangular lot is surrounded on all sides by a driveway 5yards wide. The lot is twice as long as it is wide. If thearea of the lot and driveway together is 6600 square yards,find the dimensions of the lot.

5. One leg of a given right triangle exceeds the other by 2 feet.If the hypotenuse is 10 feet, find the legs of the triangle.

6. A field of tomatoes contains 3825 plants. The numberof plantsin each row is 5 less than twice the number of rows. Find thenumber of plants in each row.

If you have satisfactorily completed your work, take the Progress test. Consult

your teacher first. After progress test 5, take the LAP TEST.

28

;

ADVANCED STUDY

I. Work any 4 of the following:

Vanatta, page 194 nos. 17-20.

page 211 no. 5; pegs 214.no. 40.

Dolciani, page 326 no. 11.

II. Work any 4.

Vanatta page 198 nos. 15, 16, 19, 20

page 214 no. 41

Dolciani page 326 no. 13.

III. Work any 4 from no. 1 and one from no. 2.

(1) Vanatta page 202 nos. 15-20

page 212 no. 18, page 214 no. 43

Dolciani page 326 no. 12, 14

(2) Dolciani page 312 no. 19, 20

IV. Work any one of these three.

(1) Graph (A) xy 36(B) xy m -10

(2) Vanatta page 205 nos. 16, 19, 20

(3) Write a report on LORAN, a system of navigation. Tellhow it-uses the concept of hyperbola (at least 500 words).

V. Work any 5 of the following.

Vanatta page 212, no. 20; page 214 no. 44, 45.

Dolciani, page 237, nos. 22, 25, 27, 28

page 309, nos. 13, 14.

APPENDIX I

I. Write the definition of conic section.

II. Name the four conic sections.

(1)

(2)

(3)

(4)

444441404

III. Write a description or definition of each of the following termsas they relate to a cone.

1. element

2. axis

3. circle

4. ellipse

5. parabola

6. hyperbola

30

APPENDIX 2

Define the following:

a. ellipse

b. foci

c. vertices

ad. major axis

a. minor axis

m

II. Identify the following parts of the ellipse in the figure:(a) major axis, (b) minor axis, (c) foci, (d) vertices (e) center.

namousesamatsermsaanaguiIVA simumemappmmirmsmommo

mosmmor.resmisimemaisasso

///. Defcribe the ellipses with the following equations:

x2(a) a2 1

x2 2

. (b) + 1a

where a.> b

where a > b

IV. In each of the following, determine (1) if the major axis is on xor y and give its length, and (2) if the minor axis is on x or yand give its length.

x2 2 x2

(a) my + 9 (b) 13* 4' n

x2 2

(c) "4f +

31

2 v,(d) 49 +9 NB 1

1

APPENDIX 2 (cont 1)

V. In each equi.tion in IV, determine the value of a and b.

a,ti

APPENDIX 3

I. Define the following:

a. hyperbola

b. transverse axis - give its leggth

c. conjugate axis - give it:: length

II. Identify the parts of the following graphl

a. conjugate axisb. transverse axisc. asymptotesd. focie. verticesf. center

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1Ail .ud.' 1011MIII. a. Write the equation of the ellipseiwrue canter is at tie origin

and transverse axis on the x axis.

b. Write the aqual:i.n of thc : xvrer ir at the or!ginand trausvers ax' . oi th y .

33

REPSRENCES 4/4741,4,

Yawata, Glen D., and Goodwin, Wilson A., Algebra Two, 4,Modern Course, Charles E. Merrill Publishing Co., 1966.

Dolciani, Mary P., Berman, Simon L., and Wooton, William,Modern Al ebra and Tri onometry, Book Two, Houghtoni n Co.,

Nichols, Eugene D., Modern Intermediate, Holt,Rinehart and Winston, Inc.. 1965.

Pearson, Helen R. and Allen, Frank n Modern Algebra,ILoicalkApacmLIBook Two, Ginn and Company, 1966.

Payne, Joseph N., Zamboni, Floyd P., Lankford, Jr.,Frances G., Algebra Two, Harcourt, Brace and World, 1969.

Dolciani, Mary P., Wooton, William, Beckenback, Edwin F.,Sharron, Sidney, Modern School Mathematics, Algebra 2,Houghton Mifflin Company, 1968.

3R Transparencies - The Circle

The Parabola

The Ellipse

The Hyperbola

Laft

EARNING

ACTIVITY

PACKAGE

1"11'444.47,6,0

SYSTEMS OF EQUATIONS

AND INEQUALITIES

mirossi

ebra 103-104MINIM

REVIEUE LAP 11,l1B1,11 3 1

iv:R.1TM BV Diane Evans

RATIONALE

Many problems in mathematics result

in mathematical models involving more than

one sentence, yet the problem requires a

single solution. Two beams of light across

the sky might each be described as the

graph of a linear equation. If they were

to cross one another, you might then have

a point of intersection, the single solu-

tion.

This LAP should help you gain an

understanding of systems of equations or

of inequalities. The use of previously

learned concepts about functions will be

used in developing methods for solving

these systems. Equations of conic sec-

tions will also be continued and studied

with systems of first and second-degree

equations so that you should be able to

solve these systems, and relate the solu-

tions to their graphs.

StCTION 1


Wthe completion of your prescribed course of study, you willbe able tos

1. Given an equation, or system of linear equations, and ordered

pairs of real numb.. determine if the ordered pairs satisfy

the equation or system of equations.

2. Given a system of two linear equations:

a. graph the system R X R

b. find the solution set of the system

3. From a system of linear equations, determine if the system is

independent, inconsistent, dependent, or consistent when given:

a. the equations of the system

b. the graph of the equations of the system

c. the solution set of the system

d. the number of ordered pairs of the solution set of the system

4. Given two systems of equations, determine if they are equivalent.

5. Given a system of two linear equations, rewrite each equation

in the form Ax + By + C 0.

6. Given a system of two linear equations, solve the system by any

one of the following methods:

a. the comparison method

b. the substitution method

c. the addition method

r,DIAON (c.Int')

Behavioral Objecrves (.:onte)

7. Given a word problem to be solved by the use of a system of two

linear equations:

a. write a system of equations which fits the problem.

b. solve the resulting system.

c. give the solution of the original problem.

RESOURCES

Ob ective 1

Nichols, read pp. 289, 290-294, Ex. 1, 3, 5, 7 page 290; 1, 3, 4page 292.

Ob1ective 2, 3

Vaiitta, read pp. 215-218, Ex. 1, 3, 4, 9, 10, 13 page 219.

Dolciani, read pp. 95, Ex. 1, 2, 5, 6, 15 page 95.

Nichols, read pp. 290-294, Ex. 1-3, 5, 7, 8 page 294.

Payne, read pp. 291-293, 303-306, Ex. 1-10 even page 294; 1-4, 6,11-13 pages 305-306.

* Appendix I

Objectives 4t_ 5

Nichols, read pp. 295, Ex. 1 a, b, 2 a, c, e pages 295-296.

Oibj ective 6

Vanatta, read pp. 219-222, Exercises

6(a) [for explanation read Nichols 296-300] Ex. 6,8,9solve by comparison.

6(b) Ex. 1, 4, 9, 10 solve by substitution

6(c) 2, 3, 3, 7 :.olve by addition

Dolciani, read pp. 9)-)8, Ex,:rcit.es

6(.0 7 2. ' pav fae: redtvg for

*Si CO,0006tD) 3, 4, 1.) prigs 9 100 solve by substitution

6(c) 6, 11, 13, 14, 17, pages 99-100 solve by addition

Nichols, read pp. 296-300, Exercises

6(a) 1-7, 9-11 pages 297-298 solve by comparison

6(h) 1-11 odd, 12 page 299 solve by substitution

6(r) 1 a,c,e,g,j,k,l; 2 a, c, e, g, h, j pages 300-301solve by addition.

Payne, read pp. 294-301, 309-310, Exercises

6(a) 2,3,4,7,8 page 298-299 (solve by comparison)

6(b) 2,4,5,6,8 page 300 (solve by substitution)

6(c) 1-12 even pages 298-299 (solve by addition)

Objective 7 (all problems are required)

* Nichols, read pp. 296-300, Ex. 3 a,c,e,g,i,k pages 301-302.

Payne, read pp. 294-301, 309-310, ex. 27, 28 pages 306.

Wooton, read pp. 210-212, Lx. 3-5, 9, 12, pages 213-214.

Dolciani, read p. 102, Ex. 6, 11, 12, 13 page 103.

* required

USE COP1 AVAILABLEObj.

1 I. Match each *trifler:ion or systom -f aquazions with the ordered pairthat belongs.LJ LioLltion set.

1. 4y -x=1

2. y 2x -I- 3

y - ' 1

3. 3x - 4y = 8

4. x + y n 6

3x + 3y = 18

A. (4,1)

B. (-5,11)

C. (7,2)

D. (-4,-5)

2 II. Graph the following systems. (Use the graph paper that follows theself - evaluation.) Determine if each system is consistent orinconsistent, dependent or independent.

5. x + y = 7

3x - 2y = 6

6. 3x - 2y = 2

6x 4y = -8

7. x 4y = 24

4x + y = 2

III. True or False.

3 8. The system 2(x - y) = 5 and 4x - 4y ms 10 is dependentand inconsistent.

9. If a system's solution set is the set of all orderedpairs, then the system is independent.

10. The solution set of the system x = 4 and y - 2 is theset containing the ordered pair (4,2).

4

5 InIN.

11. Parallel lines are independent and inconsistent.

12. A system of equations that. has only one point in itssolution ;et = 'lepenciPnt and inconsistent.

13. The fo7stcx J + v -,.. 4 i lx

3.

1"

is equivalent to y .- 2x

y- 2x = -1

con 111161ABLEtr.:ohL

4 15. The syutem x + 1 y + 2 is equivalent to x y 13(x 4. 0 a. 2(y - 1) x +lmy 1

3 45 16. Standard tom for cry 37:776 is 3y - 18 4x + 8.

6 IV. Solve the folluwina by the specified method.

COMPARISON

17. 3x + y 9x - y 16

SUBSTITUTION

19. 3x + 2y 6x 2y Is 4

ADDITION

21. 8x + 2y 1

2x + 3y = 4

7 V. Write an equation for Lach

23. The sum of two number) isnumber is 4 more than the

18. 7x + 7y 143y a X 4' 3

20. y 7x + 22x - 4y gm 5

22. 2x - 3y In 4

3x + 2y 5

of the following and solve.

59. If the sum of 12 and twice the firstsecond number, what are the numbers?

24. A collection of nicki:k arc , t..t ti rilue of $2.40 andcontains Yi coins. Rev ,Ilny of erich kind .1 1 coin are there inthe colle..ttJni

BEST COP/ MANSELVEVALUATION 1 (cont')

25. The'perimeter of a rectangle is 44 inches. If its length isdecreased twice Its width, the result is 17. Find the lengthand width.

If you have satieiactorily completed your work, take the PROGRESSTEST. Consult your teacher first.

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BehavLeral Objectives

At thu completion of your prescribed course of study, you willbe able to:

8. Given a system of a linear equation and a second-degree equation,

or a system of two-second degree equations:

a. graph the equation or equations of the system

b. solve the system algebraically

9. Given a word problem to be solved by a system of one linear

and one second-degree equation, or two second-degree equations:

a. write a system of equations which fit the problem.

b. solve the resulting system.

c. give the solution set of the original problem.

10. Graph in R X R any of the following systems:

a. a system of two linear inequalities.

b. a system of one linear inequality and one-second degree

inequality.

c. a system of two second-degree inequalities.

11. Given a graph of a system of inequalities as listed in

Objective 10, identify the solution set of that system.

RESOURCES

Objective 8

Vanatta, read pp. 231-237; ExercisesSolve by graphing: 1, 4, 5, 9 page 234 and 1, 4, 6, 7 page 237.Solve algebraically: 3, 6, 8 page 234 and 2, 3, 7 page 237.

Dolciani,SolveSolve

read pp. 320-321,by graphing: 3,5,

algebraically: 1,

Exercises7,9 page 321, and 5, 14, 15 page 325.4, 6, 9 page 321, and 4,6,14,15 page 325.

Nichols, read pp, 312-322, Exercises 1-') paste 314; 1 a,c,e,g,i,k,m,o,q and 2 a,c,e,g,i pages 316-317; 1-21 odd w-4'e 321.

I

RESOURCES (cone)

Wooton, read pp. 463 -469, ExercisesSolve by graphing: 2,6,7,9,13 page 464;Solve by algebra: 1,3,8,9 pages 466.467.

Pearson, read pp. 687-697. 698-705, ExercisesSolve by graphing 3,4,7,8,9 page 699;Solve by algebra 1,4,7,8 page 699.

ObJ ective 9 (all problems are required)

Nichols, read pp. 312-322, Ex. 2 a, c, d, page 317; 2 a, c, e page _A.

Dolciani, read pp. 320-321, Ex. 1,5,8 page 322; 34 page 325.

Wooton, read pp. 466-469. Ex. 1, 2, 4, 5 pages 467-468.

Objectives 10 11

* Vanatta, read pp. 229-230 and 238-239, Ex. 1, 3, 6, 7 page 230; 2,3, 6, 8 page 239.

Nichols, read pp. 324-327, Ex. 1 a,c,i,k,m,o,q, 2 a,c, 3 d, 4 a,c,5 a,c,e pages 327-328.

Wooton, read pp. 216-219. Ex. 19, 21, 22 page 465.

Pearson, read pp. 642 -644, 706-708, Ex. 5 EOL, 6f, 8 d,e,f page 645;1 a,b,d, 2 a,b,d,e page 708.

SELF-EVALUATION 2

Objective

8 I. Solve the following by graphing. (Use the graph paper provided)

1. x2 4. y2 m 36

x y 5

3. 4x2 - 9y2 36

2 2LC

2. y2 4x

x 2 2

4 + 9 as

4. xy 8

x + y 49 4

8 II. Solve algebraically.

S. x2 + 4y2 32x - 2y s -8

7.x2 uy'cagy+ I

Show your work.

6. x2+ y

216

2y 0 x 10

7;

2

+. 1

8. 9

x2 2.1

9

10,11 III. Solve the following by graphing. (Use the graph paper provided).

9. 2x + y < 4x - 3y > -6

10. x2 + y2 :2x y > 1

x2 2

16

11. x2 + y2 < 9 12. < 1

2a.

9>

4I

x2+ y2 49

13. x2 I 16y

3x - y : 2

SELF-EVALUATION 2 (cone) 4114Aer

14. x + Y 2

x2 2

9 -r 25

4fiffogit

9 IV. Write an equation for each of the following and solve. Showyour work.

15. The preimeter of a rectangle is 26 inches. Its area is 12 squareinches. Find the dimensions of the rectangle.

16. The product of two numbers is 8. The sum of their reciprocals is 4.What are the numbers?

17. The area of a right triangle is 24 sq. in. The measure of thehypotenuse is 10 in. Find the measure of the two legs.

18. Find two numbers such that the square of their sum is 20 morethan the square of their difference, and the difference of theirsquares is 24.

19. Find two numbers whose difference is 2 and whose product is 2.

SELF-EVALUATION 2 (Cont')

20. Find two numbers such that the sum of their squares is 170 and thedifference of their squares is 72.

If you have satisfactorily completed your work take the LAP Tur,Consult your teacher first.

13

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aSELF-EV PlUATION 2 (cortt I)

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Objective UWE

APPENDIX 1

3c I. Given the following aolutionn to the systems of linear equations,determine if each system is inconsistent, consistent, dependent,or independent.

a. (3,8) in common

b. no points in common

c. all points in common

d. (-3,2) in common

a. equations are parallel

f. equations have same graph

3d II. Given the following numbers of ordered pairs of solution setsof linear equations, determine if each system is inconsistent,consistent, dependent, or independent.

a. no points are in the solution set

b. one point is in the solution set

c. an infinite number of points are in the solution set

11471:04Aiwa

ADVANCED STUDY

I. Dolciani, read pp. 100A to 101, ex. 1-6, 12, 13, 14, 15 page 101.

II. Work ary one of the following:

1. Nichols, read pp. 302-310, ex. 1, 2 page 306;1, 2a,c,d,e,h,j,l,m,o pages 310-311.

2. Payne, read pp. 327-329, 334-336, ex. 1-10 page 329; 21 page337.

3. Wooton, read pp. 649-452, 657-661, ex. 1-12, 21 -26 pages 652-653; 1-8 page 661.

Dolciani, ex. 11-14 page 326.

III. Wooton, read pp. 223-226, ex. 1-20 oral page 226.

IV. Vanatta, read pp. 224-227, ex, 1-4, 6 pages 227-228.

REFERENCES


Vanatta, Glen D, and Goodwin, Wilson A., Algebra Two, 'A 'ModernCourse, Charles E. Merrill Publishing Co., 1966.


Dolciani, Mary P., Berman, Simon L., and Wooton, William, ModernAlgebra, Book Two, Houghton Mifflin Co., 1965.


Nichols, Eugene D., Modern Intermediate Algebra,, Holt, Rinehartand Winston, Inc., 1965.


Pearson, Helen R., and Allen, Frank B., Modern Algebra: A LogicalApproach, Book Two, Ginn and Company, 1966.


Payne, Joseph N. Zamboni, FLoyd F., Lankford, Jr., Francis G.,Allard Ts, Harcourt, Brace and World, 1969.


Dolciani, Mary P., Wooton, William, Beckenbach, Edwin F., Sharron,Sidney, Modern School Mathematics, Algebra 2, Houghton MifflinCompany, 1968.

L

1.

EhRNING

ACT IV ITV

Pa CKAGE

A

ca+c,b+dir--zi, 412

>8I

COMPLEX NUMBERS

REVIP:.!ED BY

A I .ebra 103- 10/I

LAP NUMBER 32

ITT FN BY Diane Evans

4

41,%;11,NAlf

Some polynomial otimitions hsvo no real-roots.

1:01 eXiswp1'.!. N 1 - 0 br

there is no real numbar which, when squared,

he negative. ThureforP, we construct a number syst,:!i;

which cootains ''or all equationn.

whir.h it caii the cowp1eA ruior sy,,,tery contains :.h(2

real numbers as a subset; satt.fies the eleven f.k,10

postulates; imd CO.hiriS a solution tar all polynomial

equations with (!i1,; coefficients.

This 1AP presents the coAplox numbers as an exten-

silo of the real numbers and defines the41 as order:A

pairs of real numbers. Some reasons for the ordered

pair approach are:

(1) Graphs of complex numb2rs seem more naturai.

.2) It leads easily dnd naturaily to Ue Moivre'.3Theorem,

(3) it can bo ilz;n0 later to sho,a the connectionbetween vc:..crs, c000ltm numhcrs, poldrcoordinati!i, and Point t-, in a olano.

The ...tandard i'vtarion, ri f hi, 1(: 1.1troduc-bd -0

thdt students abie to compute wirh tnis

dS well as with the ,Jrder.!d pair torn.

3


After having completed your prescribed course of study, you will beable to:

1. Define imaginary number, complex number, pure imaginary number.

2. Simplify any imaginary number.

3. Given the sets of numbers natural, whole, integer, rational,irrational, veal, and complex, determine if relationships amongthese sets are true or false.

SECTION 1COPI AVAILABLE

4. Given two complex numbers in ordered pair form, determine ifthey are equal.

5. GivPn twn equal ordered palm in which one or more of thecomponents are variables, determine the value of each variableaccording to the definition of equality of complex numbers.

6. Giveo two complex numbers in order pair form, give the orderedpair that names:

a. their sumb. the additive inverse of either numberc. their difference

7. Determine if the following properties apply to the set of complexnumbers and/or identify examples of each:

a. closure under additionb. commutative for additionc. associative for additiond. additive identitye. additive inverses

8. Given two complex numbers in ordered pair form, give the orderedpair that names:

a. their productb. the multiplicative inverse of either number provided the

number is not zeroc. their quotient, if it io definedd. the nth power of either

9. Determine if the following properties apply to the set ofcomplex numbers and/cr identify examples of each:

a. closure for multiplicationh. commutative for multiplicationc. associative for multiplicationd. the multiplicative identitye, multiplicative inversesf. distributive

2

SECTION 1 (cont')

Behavioral Objectives (cont.)

40104,41444

10. Given a number system, determine whether or not it is a field.If it is not, list the missing properties.

21. Given any complex number, express it in the form a + bi wherea and b are real numbers.

12. Given two equal complex numbers in which one or more of thecomponents are variables, determine the value of each variableaccording to the definition of equality of complex numbers.

13. Given two complex numbers, expressed in standard form, compute:

a. their sumb. the additive inverse of either numberc. their differenced. their producte. the multiplicative inverse, provided the number is nonzerof. their quotient, if it is defined

14. Evaluate any expression of the form is where a is a naturalnumber.

3

SECTION 1

RESOURCES

Ctjective 1

Vanatta, read pages 350-353, Ex. -define the terms in the objective.

Objective 2

Vanatta, read pages 350-352, Ex. 1-14 page 352.

Objective 3

Vanatta, read paves 350-353, Ex. - Appendix I.*

Objective 4, 5

Nichols, read pages 251-252; Ex. 1 a, c, d, e, g, i, j; 2 pages 252-253.

Objective 6, 7

Nichols, read pages 253-258; Ex. 1-26 even pages 253-254; 1-4 page255; 1-3 page 256; 1, 2, 3 a, c, e-q, 4 page 258.

*Appendix II - Part I

Objective 8, 9

%ichols, read pages 258-266; Ex. 1, 2 page 259; 1, 4, 5, 6 pages 260-261; 1, 2 pages 263-264; 1, 2 a, c, e-t, 3, 4 a, b pages 266-267.

*Appendix II - Part II

Objective 10

Nichols, read pages 267-268; Ex. 1-5 page 268; 1-24 even pages 268-269.

*Appendix III

Objectives 11, 12, 13, 14

Nichols, read pages 269-270; Ex. 1 a, c, e, 2, 3 a, c, e, 4 a, c, e,5, 6, 7, 8, 9 a, c, e pages 270-273.

Payne, read pages 24-41, Ex. 1-33 even page 26; 1-32 even pages 28-29;1-16 pages 30-31; 1-27 even pages 33-34; 1-25 even page 36; 1-15page 38; 11-31 page 41.

Wooton, read pages 364-374; Ex. 1-24 page 367; 1-50 odd pages 368-369;1-42 even pages 373-374.

Pearson, read pages 591-603; Ex. 1-7 pages 593-594; 1-13 pages 597-598;1-6 pages 601-602; 1-4, 6-10 page 604.

* required 4

SELF-EVALUATION 1

OBJ.1 I. WRITE THE DEFINITION OF THE FOLLOWING:

a. imaginary number

b. complex number

c. pure imaginary number

2 II. SIMPLIFY THE FOLLOWING:

a) 470.

b)

c) -475.

d) 4.717

3 III. FOR THE FOLLOWING USE N = NATURALS, W = WHOLES, I = INTEGERS,Q = RATIONALS, T = IRRATIONALS, R = REALS, and C = COMPLEX TOANSWER TRUE OR FALSE.

a) N c T

b) T c C

c) Rc C

d)W=R

e) Q T

f) T c C

g) I c Q

4,5 IV. FOR WHAT VALUE OF x AND y IS EACH OF THE FOLLOWING STATEMENTS TRUEWHEN x, y e Real s?

a) (x, -3) = (2, y)

b) (-x, 6) = (2, 2y)

c) (2 + 31,4) = (5, + 3)

d) (3 - m, n - 3) = (6, 6)

6 V. COMPUTE.

a) (3, -4) (6, -6)

5

SELF - EVALUATION 1 (cent')

b) (a, -b) + (-s, b)

c) (3, 2) - (4, 9)

d) (c, 0) - (do 0)

e) (2, -4) + (8, 3)

6 VI. GIVEN THE ADDITIVE INVERSE OF THE FOLLOWING.

a) (3, 4)

b) (-2, -6)

c) (-2, 2)

d) (8, -10)

e) (0, 0)

8 VII. COMPUTE.

a) (7, 4) (3, -4)

b) (-2, 6) (3, -1)

c) (6, 0) (0, 6)

d) (1, 0) (b, 0)

e) (-10, 7) (2, -3)

f) (4. 3) + (2, 3)

g) (10, 2) (2, 3)

h) (a, b) (-c, -d)

1) (2, 8) (8, 8)

8 VIII. GIVE THE MULTIPLICATIVE INVERSE OF THE FOLLOWING:

a) (3, -5)

b) (- , 0)

c) (5, 0)

d) (0, 0

EST CD& AVAILABLESELF-EVALUATION 1 (cont')

8 IX. EVALUATE AND EXPRESS THE RESULT AS AN ORDERED PAIR.

a) (0, 1)4

b) (0, 1) 7

c) (0, 1)2

11 X. EXPRESS THE FOLLOWING COMPLEX NUMBERS IN STANDARD FORM. (a = bi,where a,b c Reals.

a) (3, 5)

b) (0, -2)

c) (a, b)

12 XI. FO% WHAT VALUE OF x and y IS EACH OF THE FOLLOWING STATEMENTS TRUEWHEN x, y e Reals?

a) x + 31 = 5 + y1

b) x + yi = 4i

c) x + yi = 7

13 XII. EXPRESS THE FOLLOWING IN THE FORM a + bi when a, b c Reals.

a) (3/2. + 51) + (4. - 71)

b) (-5 - 3i) - (2 - 6i)

c) - 061 + 0

d) (2 + 31). (5 + 41)

e) (-5 - 31) - (2 - 6i)

f) -(6 - 5i)

gi 771:7

h) (3 + 50(2 + 41)

i) (8 + 2i) + (6 + 4i)

j) (5 + 61) (2 + 3i)

14 XIII. EXPRESS THE FOLLOWING IN THE FORM a + bi where a, b c Reals.

a) i2

7

SELF - EVALUATION 1 (cent') 4141Vatv

-.4414t4kr

7, 9 XIV. NAME THE PROPERTY ILLUSTRATED BY EACH OF THE FOLLOWING AND STATEWHETHER OR NOT EACH HOLDS FOR THE COMPLEX NUMBERS. z1- 22, and z

sARE THREE COMPLEX NUMBERS.

a) zi + z2 = z2 + zi

b) zi + (0, 0) = zi

1

c) z3 = (1, 0)

d) 21 + (z2

+ Z3) = (2

1+

2) + Z

s

e) z1

(z2+

3) = (21 2

2) + (2

123 )

0 21 + 22 = (0, 0)

10 XV. STATE IF EACH OF THE FOLLOWING IS TRUE OR FALSE. IF FALSE, STATETHE PROPERTY(S) TO MAKE THE STATEMENT TRUE.

a) The set of integers is a field.

b) The set of real numbers is a field.

c) The set of irrational numbers is a field. /,d) The set of natural numbers is a field.

e) The set of complex numbers is a field.

IF YOU HAVE SATISFACTORILY COMPLETED YOUR WORK, TAKE THE PROGRESS

TEST. CONSULT YOUR TEACHER FIRST.

8

StLIION


After having completed your prescribed course of study, you will beable to:

15. Compute the absolute value of a complex number.

16. Determine the conjugate of a complex number.

17. Given a pair of complex numbers:

a. Plot the complex numbers in the coordinate plane.b. Plot the additive inverse of either complex number in

the coordinate plane.c. Plot the conjugate of either complex number in the

coordinate plane.d. Draw on the plane a geometric interpretation of the sum of

the two complex numbers.e. Draw on the plane a geometric interpretation of the differ-

ence of the two complex numbers.

18. Given an expression involving square roots of real numbers,write the expression in simplified form so that the radicand isa positive integer.

19. Given any quadratic equation over the real numbers, determineits solution set in the complex numbers.

20. Given a set of two complex numbers, write a quadratic equationwith real coefficients having this set as its solution set inthe field of complex numbers.

RESOURCES

Objectives 15, 16

Nichols, read pages 273-274; Ex. 1-4 pages 273-274; 1-5 pages :14-275.Payne, read pages 38-40, 43-44; Ex. 1-10 page 41; 1-19 page 45.Pearson, read pages 603-604, 610-615, 617-619; Ex. 5 a-n, 11, 12, 13

pages 604 -605; 3 page 615; 8, 11 page 616; 1-5, 7, 9 pages 619-620.

Objective 17

*Nichols, read pares 275-278; Ex. 1 a, c, e, 2 page 276; 3 a-e pp. 277-278; 1-3 pp.278-279.

Objective 18

Nichols, read pages 279-280; Ex. 1-2 pages 280-281.Payne, read pages 24-29, Ex. 1-33 even pages 28-29.Wooton, read pages 365-369; Ex. 1-24 page 367; 1-50 even pp. 368-369.Pearson, read pp. 605-608; Ex. 1, 2 page 608.

Objectives 19, 20

Nichols, read page 281-283; Ex.

Payne, read pp. 254-255, 269-2711-18 page 271.

Wooton, read pages 374-376; !ix.Paarcon, road naloc, A05 -AnR; Fx.

1, 2 page 284.

; Ex. 3, 8, 12, 16, 25-30, 40-47 p. 256;

1-26 page 377.3-5, 7, R nana Ana,

SELFEVALUATION 2

OBJ.16, 16 I. FIND THE ABSOLUTE VALUE AND CONJUGATE OF THE FOLLOWING COMPLEX

NUMBERS.

a) 10+ 3i

b) -3i

c) 9

d) 5 - 2i

e) - i

17 II. A. ON THE GRAPH BELOW. PLOT THE FOLLOWING COMPLEX NUMBERS.

2) 2 + 6i

3) 5i

4) -6

--I--

-1-1-

_1_

-.:-.21:6

--1-1--

-1-4

I

77--......

i

--::-1----/--

0, 111,

..."1--,

...0.1--S 1

.4

-3 2 - ItifI g 3 1

i

......

7

........

._L_

±I..

...

L it._..1

B. PLOT THE ADDITIVE INVERSE OF EACH COMPLEX NUMBER IN EXERCISE A.

_III; 17E- __, 1_1

--1-8i 4

- -.--. -- 1t

I,

r

.. 3

--I -- -6

1----1---.-

-4 .-4 -3 .-2

-i---1

1

:2

-.,5

-JI 2 3 4 6

I iG I

t

1 1--1

- --,

1

--I

iI

10


C. PLOT THE CONJUGATE OF EACH COMPLEX NUMBER IN EXERCISE A.--7--1

1M..

......

.... ....

...

.....1 . .1 ,.... ... 4.,

.

k

.....

-

.....

.a.

....

-...

,4...4 ,.......

IONO Or

±x.3

'...:3,1

**6'.1171

p+

171...

''...'"1...1-A.n.

- . . . . t, -s--I ...2 1

..4

; 3 4 6NW

64.1Ir

i..L.....1.....1....

1

....I

M21 .... ..........

[.

MOM=

I gilla

MN iD. CONSTRUCT A GRAPH FOR EACH OF THE FOLLOWING:

(1) (-2, 4)4. (3, 2) (2) (-2 + 61) - (4 - 21)

lr..,IMIIMI

NI.

4. OM.,

I ....

....,

r.

4....4

. 3

4

.... ...o.

....o

00

am a................

0.11t;It......... ....f....... . ..

11

1,.I.....0 --- x

MPG

'"1.4 4 ..3 ..

1

s 3

L........."4 I

........1 ...4

...3

4

...........

5

........ ......

....

(3) (0, 6i.

_ 6

) + (0, 41)

1. JJt- =.lo ;

v't

I 11-.LS I

1-1-

1--...i

7!

.1...i -1G

r-----T---

4

.... 3

2

I

.1.1. riffs./7AM.

...........NM,

*

...

....Et "4

0

...........

IA....6 5 4 .-3

'''''t ..`2

4

.I:

1 2 3....

4 6.....i....1,....7"'"I'''f71.,

0_11E1

i

TI

iL .

1

1 t... _

I 7Y I

(4) (-3 - 41) - (5 - 21)

17i -1771-;

f -; 1-r- -r---t---T T rr 4

H1 i 1

r--1--i-T-7---x . i 1 1 i7 1 L 1 3 2

I 1 I I

-L. .

, I I

- 41. "

SELF-EVALUATION 2 (cant')

18 III. SIMPLIFY EACH OF THE FOLLOWING:

A) +

B) 72' -

C) -41" /7Mk D)

01:71 + 42/f

E) sci 47F) -47 -

19 IV. FIND THE SOLUTION SET OVER THE COMPLEX NUMBERS.

A. x2 + 9 = 0

B. zx2 + 10 = 4x

C. 3x2 + 12 = 0

20 V. FIND A QUADRATIC EQUATION OVER THE REAL NUMBERS HAVING THE GIVENSET AS ITS SOLUTION SET.

A) (-2 + 5i, -2 - 51}

B) (is -1)

IF YOU HAVE SATISFACTORILY COMPLETED YOUR WORK, TAKE THE LAP TEST.

CONSULT YOUR TEACHER FIRST.

MC MEIMaiAPPENDIX I

Using the following Venn Diagram of the set of complex numbers,determine if the given statements are true or false.

1. (Z)= C

2. T a N 9. W=C

3.

4.

Nc W

R C

10.

11.

TICW c T

5. Na C 12. W Q

6. Wa I 13. Ra N

7. Q= R 14. I C

8. N= Q 15. Ca T

APPENDIX II

PART I

Objective 4

Determine if each of the following properties apply to the set ofcomplex numbers. Give an example for each.

a) closure for addition

b) commutative for addition

c) associative for addition

d) additive identity

e) additive inverse

PART II

Objective 6

Determine if each of the following properties apply to the set ofcomplex numbers. Give an example for each.

a) closure for multiplication

b) commutative for multiplication

c) associative for multiplication

d) multiplicative identity

e) multiplicative inverses

f) distributive

14

BEST COP! AVAILABLE

APPENDIX III

Write an X by each property that holds for the given set.Write a circle (0) by each Fropetty that do's not hold.Do not leave a blank.

PROPERTIES NATURAL WHOLE INTEGERS RATIONALS IRRATIONALS REALS dom kx

Closure. for +

Closure for x

Commutativefor +

Ccmmatativefor x

Associativefor +

Associativefor x

Distributive

Additiveidentity

Additiveinverse

Multiplica-tive identi-ty

Multiplica-tive inver-ses

15

REFERENCES

1. Vanatta, Glen D., Al ebra Two: A Modern Course

Charles E. Merrill Pub is ing Co., 1966.

2. Dolciani, Mary P., Berman, Simon L., and Wooton,

William, Modern Algebra, Book Two, Houghton

Mifflin Co,,

3. Nichols, Eugene D., Modern Intermediate Algebra,

Holt, Rinehart, and Winston, Inc., 1965.

4. Dolciani, Mary P., Wooton, William, Beckenbach,

Edwin F., Sharron, Sidney, Modern School Mathe-

matics Algebra Houghton Mifflin Co., 1968.

5. Payne, Joseph H., Zamboni, Floyd F., Lankford,

Jr., Francis G., Algebra Two, Harcourt, Brace and

World, 1969.

6. Pearson, Helen R., and Allen, Frank B., Modern

agatmajfusaisAi Approach, Book Two, Ginn and

Company, 1966.

16

EARNING

ACTIVITY

PAC KALE

INTRODUCTIONTO PROBABILITY

REVIEWED BY 140 NUMBER Na 36

WRITITN BY Mrs, Diane Evans

31273.11

RATIONALE

PROBABILITY

Mathematics has long been closely associated

with the physical sciences. Recently, there have

been important applications of mathematics to the

social and biological sciences, and application to

all sciences have increasingly involved probability

and statistics. The increasing significance of pro-

bability and statistics throughout our society makes

it more and more important that people gain knowl-

edge of probability and statistics.

This LAP takes a brief introductory look at

probability. You will find this study of probabil-

ity not only fun in itself, but also the basis for

beginning the study of statistics.

NOTE: This LAP does not have any tests. There arethree assignments that you are to completeand turn in to the teacher. You will begraded on these assignments.

1

SAMPLE SPACES

So far in your study of mathematics, the concept of "set" has

played an important role. A particular kind of set very important in

the study of probability, consists of a collection of all the possible

outcomes of an experiment. This set is called a sample space.

Example It Suppose you are asked to toss two coins, a dime and a quarter.

In how many ways can they land?

Solution: Here is a list of all the possible outcomes of the toss:

Dime Quarter

H H Both heads

Dime, heads; quarter, tails

Dime, tails; quarter, heads

Both tails

The sample space for the above experiment may be written

{HH, HT, TH, TT}. In this set, the first letter in each pair represents

the outcome of the toss of the dime, the second letter represents the

quarter.

Example 2: Consider the experiment where three different coins are tossed:

a dime, quarter, and a penny. The list of all possible out-

comes appears as follows:

Dime Quarter Penny

H H H

H H T

H T H

H T T

T H H

T H T

T T H

T T T

2

IIIMAMOULABLE

Again, the list of allypossible outcomes or the sample space is as

follows: {HHH, HHT, HTH, HIT, MN, THT, TTH, TTT }.

Example 3: Suppose you are asked to toss a die. What is the sample space

of this experiment?

Assuming the die is conventional and your toss is legiti-

mate, it can land,with only six faces up. Therefore, the sample

space for this experiment is the following set: (1, 2, 3, 4, 5,

6 }.

Example 4: Consider the problem of tossing a pair of dice. What is the

sample space of this experiment?

To clarify the experiment in our minds, let us call one

die, die (1) and the other die, die (2). Thus an outcome of

a three on die (1) and a four on die (2) is different from an

outcome of a four on die (1) and a three on die (2). In deter-

mining the sample space of this experiment, the following

chart may be helpful:

Outcomes of Die (2

1 2

(1,1 1 2 1 3 1,4 1 5 1 6__1

2 21 111111M1 ( 2 4 25 26

31 32 33 34 35 36

4 41 42 43 44 45 (46

l_____i5 1) 5,2 5 3 5 4) 5,5) (516)

6 61 6,2 63 64 66 66

The outcomes for die (2) are listed on top of the chart,

while the outcomes for die (1) are listed along the left.

The entries of the table are to be interpreted as follows:

3

(2,3): 2 on die (1) and 3 on die (2)

(3,2): 3 on die (1) and 2 on die (2)

ASSIGNMENT ONE

Answer the following questions and TURN THEM IN to the teacher.

I. List a sample space for each of the following experiments:

1. A die is tossed, then a coin is tossed.

2. A three-digit numeral is to be formed from the digits 2, 4, and5, with no digit to be used more than once in a numeral.

3. A box contains four balls: one white, one red, one blue, and

one green. You are to draw one ball from the box, replace it

and then draw a second ball.

4. Repeat exercise 3, except this time you draw two balls from the

box, one at a time, without replacement.

5. Four dimes are tossed.

II. Using tne sample space for die (1) and die (2) on Page 3, answerthese questions:

6. How many of the outcomes show both the dice with the same outcome?For example: (1,1) etc.

7. ,Now many outcomes have the sum of 7?

8. List the outcomes when die (1) is 1.

9. List the outcomes when die (2) is I.

10. Using the results of problems 8 and 9, list the intersection and

union of the two sets of outcomes.

PROBABILITY

The word "chance" is often used in conversation. We may say, "we

take a chance on a raffle," "the chances are the Gamecocks will win,"

"the chances are we will not get a test today."

Sometimes W2 use numbers when talking about "chance." For example:

4

,14:i41"1" In this section we will give the notion of chance a definite mathe-

matical

chances are 2 to 1 out team will win."

lr "The chances are 50-50 that a coin will come up tails."

40PFmatical meaning. We will do this by introducing the concept of probability.

Let's consider an appropriate scale for probability. Study the chart

below which shows a scale starting at 0 and extending to 1, with .5 repre-

senting the half-way point.

SCALE OF PROBABILITY..1111M1111111.11.4.1111i0111.1

Probability objects will roll downhill

Probabi 1 i iy I wi i I die someday

ProbabiliLy our team will win --

Probability it will rain 1--ylay

Probability a coin wil; land Heads

Probability I wi!1 get an A --------

Probability I will live to 10C

Probability the coin will land onits edge

Probability of getting a 7 on t4,.toss of an ordinary di?

certain 1.00

.5

impossible 0

The top end represents an event we know will happen for certain.

The bottom of the scale represents an event that is an impossibility.

For example: gettina d 7 on a die that has only six faces numbered 1

to 6 has a probahlity of zero. Any event which is impossible to occur

.N will always have a probability of zero, Some events are certainties.

For example: the probahility that I will die some day is a certainty.

Any event which is a certaioty wi!1 have a probability of one. Some

events are easy to place on t': :air; ot!!erc,, however, cause us to

wonder.

.54

4 need to determine hoe We can arrive at the measure of the probe

bility of a real event when it's neither a certainty nor an impossibility.

EQUALLY LIKELY EVENTS

Consider the experiment of tossing a coin. Since there are two

possible outcomes, a sample space for this experiment consists of just

.these two outcomes: (H, T}.

It seems reasonable to assume that heads and tails both have an

equal opportunity to occur. When each element in the sample space has

an equal chance of occurring, we say they are equally likely tc happen.

Definition: Probability of an event

If an experiment can result in N different outcomes, and if each of

these outcomes is equally likely, and if m of these outcomes corresponds

to event A, the probability of event A, written P(A) is:

P(A) a I

In other words:

The probability of Athe number of favorable outcomes

=Total possible outcomes

Example 1: Let's consider again the experiment of flipping a coin, N = 2,

since there are two different outcomes, heads or tails.

Therefore, the probability of the event heads is and the

probability of the event tails is 1/2; written:

P(H) = z P(T) - 1/2

Consider the following experiment. Do you think this experiment

verifies the result of example one?

During World War II, one mathematician in an interrment camp

tossed a coin 1,000 times and then repeated this experiment

over and over. He kept a record of the results. In ten sets

of 1,000 tosses, he found that the number of heads was: 502,

6

511, 497, 529, 504, 476, 507, 5214 504, and 5g NottO how

the numbers of heads clustered around 500 (one-half of 1,000)

althouth.none of the numbers was precisely 500. It is impor-

tant to examine the results of Stith experiments to find if

they do agree with our definition.

Spin the penny: Do you think the results will be the same if

you spina'penny as they are when you toss a penny?

Take a brand new penny--be sure it is new--and spin it so that

it rotates for a large number of turns. Observe the results.

Repeat the spin 50 times and record the number of heads. Do

the results approximate 1/2?

Example 2.1. Consider tossing a die. The six faces are marked by dots as

it :1

shown. There are six possible equally likely outcomes. Our

2, 3, 4, 5, 6). Therefore,

Since 7 is not a member of the sample space, P(7) = 0.

Every element in the sample space is called an elemental./

event.

Notice, each outcome of the experiment corresponds to a single

element in the sample space.

11Isample space looks like this: (1,

applying our definition41I11S aniP(1) = P(2) n P(3) =

P(4) = P(5) = P(6) =

Example 3: We toss a die again as in example one. This time we are con-

cerned as to whether' the outcome is an even number or an odd

number. What is the probability the outcome is even?

Solution: The event "outcome is even," consists of the three elementary

events (2, 4, 6). Since three of the six equally likely events

7

3

result in even numbers, we say-- P(even) = 6 or h.

Example 4: A class of 30 students selects its class president in the

following way. The name of each student is written on a

slip of paper and placed in a box. The, slips are mixed up

by shaking the box. Without looking, a person draws a slip.

The person named on the slip is appointed class president.

Questions:

(A) How many elements are in the sample space?

(B) What is the probability of a particular individual

being chosen president?

Solution: For A: Since there are 30 students in the class, there are

30 elements in the sample space (or 30 possible outcomes).

For B: Since there are 30 different outcomes, P(A particular

person is chosen) = -31-1

Example 5: In a bag there are 7 balls, 3 balls are red and 4 balls are

green.

a) What is the probability of picking a red ball on the first

draw?

b) What is the probability of picking a green ball on thesecond draw, given that a red ball was picked on the first

draw?

c) What is the probability of picking a red ball on the third

draw, given that two green balls have been picked on thefirst and second draws?

Solution: a) Sincere there are seven balls and three of them correspond

to choosing a red ball, the probability of choosing a red

ball is

b) Since a red ball was drawn on the first draw, we have six

balls left, two red balls and four green balls. AThergfore,

the probability of choosing a green ball is now it- or 1 .

c) Since two green balls have been drawn, there are five balls

left, three red balls and two green balls. Therefore, the

probability of choosing a red ball is .

8

ASSIGNMENT TWO

Answer the following questions and turn them in to the teacher.

1. In a bag there are six white balls and one red ball. What is theprobability of picking a red ball on the first draw?

2. In a bag there are six white balls and two red balls. What is theprobability that you will pick a white ball on the first draw?

3. What is the probability you will get a three when you toss a dle?

4. What is the probability you will not get a three when you toss adie?

5. A pupil is required to read a play, a short story, and a poem.What is the probability that he reads the poem first (assumingthat each is equally likely to be in the book he picked up first)?

6. If the batting average of the star shortstop is .289, what is theprobability he will get a hit the next time he is at bat?

7. If you have one dime and seven pennies in your pocket, what is theprobability that the coin you take out at random will be a penny?A dime? A nickel?

ASSIGNMENT THREE

Answer the following questions and TURN THEM IN TO THE TEACHER.

1. The probability of getting a head when tossing a coin is 1/2. Toss acoin 30 times. Give the number of times heads come up.

2. Suppose you toss a red die and a green die at the same time,

a. List the sample space.b. Give the probability of rolling a 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

and 12.c. What is the probability of rolling a 1 when rolling two dice?d. Roll two dice 30 times and record the results to show how close

to true probability each result !s.

3. If you choose a card from a regular br'dge deck,

a. What is the probability you will choose a 10?

9

b. What is the probability you will choose a heart?c. What is the probability you will choose a 10 of hearts?

Use the Time-Life book Mathematics, pages 134-147, to answer 4 and 5.

4. When playing poker, what is the probability of getting a Royal Flush

(10, Jack, Queen, King, Ace, of the same suit)?

5. What is the probability of a family having 13 boys and no girls?

6. Read one of the following and write a report (at least 50 words).

A) Chapter 3 from Probability and Statistics for Ever oneby Irving Adler.

B) Any chapter that interests you from How to Take a Chance

by Darrell Huff and Irving Geis.

REFERENCES

How to Take a Chance, Darrell Huff and Irving Geis,W. W. Norton and Co., Inc., 1959.

Probability and Statistics for Everry Man, IrvingAdler, The John Day Co., 1963.

Mathematics (510), David Bergamini, by the editorsof Time-Life Inc., 1963.

to

10

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